Question: Consider the points $\left(x_{1}, y_{1}\right)=(-1,0),\left(x_{2}, y_{2}\right)=(0,3),\left(x_{3}, y_{3}\right)=(1,3),\left(x_{4}, y_{4}\right)=(2,5)$, $\left(x_{5}, y_{5}\right)=(3,9)$. Find the least squares line for these points.

Answer: $y=2+2 x$.

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Question:
In $\mathbb{R}^{4}$ with the usual dot product, find an orthogonal basis for
$$
\operatorname{span}\left\{\left[\begin{array}{l}
0 \\
1 \\
1 \\
0
\end{array}\right],\left[\begin{array}{c}
3 \\
0 \\
2 \\
-1
\end{array}\right]\right\}
$$

Answer: $\mathbf{u}_{1}=\left[\begin{array}{l}0 \\ 1 \\ 1 \\ 0\end{array}\right], \mathbf{u}_{2}=\left[\begin{array}{c}3 \\ -1 \\ 1 \\ -1\end{array}\right]$

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Question:
Determine if the system is consistent. If so, is the solution unique?
$$
\begin{gathered}
x+2 y+z-w=2 \\
x-y+z+w=0 \\
2 x+y-z=1 \\
4 x+2 y+z=3
\end{gathered}
$$

Answer: The solution is: $x=\frac{1}{3}-\frac{1}{3} t, y=\frac{2}{3}+\frac{2}{3} t, z=\frac{1}{3}, w=t$.

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Question: Let $\mathbf{u}, \mathbf{v}, \mathbf{w}$ be linearly independent vectors in $\mathbb{R}^{n}$. Are the vectors $\mathbf{u}+\mathbf{v}, \mathbf{u}+\mathbf{w}$, and $\mathbf{w}+\mathbf{v}$ linearly independent?

Answer: From $a(\mathbf{u}+\mathbf{v})+b(\mathbf{u}+\mathbf{w})+c(\mathbf{w}+\mathbf{v})=\mathbf{0}$ we get $(a+b) \mathbf{u}+(a+c) \mathbf{v}+(b+c) \mathbf{w}=\mathbf{0}$. Since $\mathbf{u}, \mathbf{v}, \mathbf{w}$ are linearly independent, this last system has only the trivial solution, so $a+b=0, a+c=0$, and $b+c=0$. Solving, we find the unique solution $(a, b, c)=(0,0,0)$. So the vectors $\mathbf{u}+\mathbf{v}, \mathbf{u}+\mathbf{w}$, and $\mathbf{w}+\mathbf{v}$ are linearly independent.

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Question: Solve the system of equations $8 x+2 y+3 z=-3,8 x+3 y+3 z=-1$, and $4 x+y+3 z=-9$.

Answer: Solution is: $[x=1, y=2, z=-5]$.

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Question: Find the area of the triangle determined by the three points, $(1,2,3),(2,3,4)$ and $(3,4,5)$. Did something interesting happen here? What does it mean geometrically?

Answer: Let $P=(1,2,3), Q=(2,3,4)$, and $R=(3,4,5) . \overrightarrow{P Q} \times \overrightarrow{P R}=\left[\begin{array}{l}1 \\ 1 \\ 1\end{array}\right] \times\left[\begin{array}{l}2 \\ 2 \\ 2\end{array}\right]=\left[\begin{array}{l}0 \\ 0 \\ 0\end{array}\right]$. The area of the triangle is 0. It means the three points are on a line.

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Question: Find $\left[\begin{array}{l}1 \\ 2 \\ 3 \\ 4\end{array}\right] \cdot\left[\begin{array}{l}2 \\ 0 \\ 1 \\ 3\end{array}\right]$.

Answer: $\left[\begin{array}{l}1 \\ 2 \\ 3 \\ 4\end{array}\right] \cdot\left[\begin{array}{l}2 \\ 0 \\ 1 \\ 3\end{array}\right]=17$

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Question:
Consider the following augmented matrix in which $*$ denotes an arbitrary number and $\mathbf{I}$ denotes a non-zero number. Determine whether the given augmented matrix is consistent. If consistent, is the solution unique?
$$
\left[\begin{array}{ccccc|c}
\mathbf{\square} & * & * & * & * & * \\
0 & \mathbf{\square} & * & * & 0 & * \\
0 & 0 & 0 & 0 & \mathbf{\square} & 0 \\
0 & 0 & 0 & 0 & * & \mathbf{\square}
\end{array}\right]
$$

Answer: There might be a solution. If so, there are infinitely many.

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Question:
On $C[-1,1]$, which of the following functions are orthogonal to each other?
$$
f_{1}(x)=x, \quad f_{2}(x)=x^{2}, \quad f_{3}(x)=x^{3}-x, \quad f_{4}(x)=1-x^{4} .
$$

Answer: $f_{1} \perp f_{2}, f_{1} \perp f_{4}, f_{2} \perp f_{3}$, and $f_{3} \perp f_{4}$.

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Question:
Find the least squares approximation for the system of equations
$$
\begin{array}{rr}
x+2 y+2 z= & 5, \\
x+y-z= & 11, \\
x+2 y-z= & -18 \\
2 x-y+2 z= & 0 .
\end{array}
$$

Answer: $(x, y, z)=(-1,-1,2)$.

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Question:
Find the least squares approximation for the system of equations
$$
\left[\begin{array}{rrr}
-1 & 2 & 1 \\
-1 & 0 & -1 \\
2 & 0 & 2 \\
0 & 0 & 2 \\
1 & 2 & 2
\end{array}\right]\left[\begin{array}{l}
x \\
y \\
z
\end{array}\right]=\left[\begin{array}{r}
1 \\
1 \\
3 \\
-2 \\
4
\end{array}\right]
$$

Answer: $(x, y, z)=(2,2,-1)$.

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Question: Solve the system of equations $3 x-y+4 z=6, y+8 z=0$, and $-2 x+y=-4$.

Answer: Solution is: $[x=2-4 t, y=-8 t, z=t]$.

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Question: Find the principal axes of the ellipsoid $2 x^{2}+2 y^{2}+3 z^{2}+2 x z-2 y z$.

Answer: The principal axes are $\frac{1}{\sqrt{3}}\left[\begin{array}{r}1 \\ -1 \\ -1\end{array}\right], \frac{1}{\sqrt{2}}\left[\begin{array}{l}1 \\ 1 \\ 0\end{array}\right], \frac{1}{\sqrt{6}}\left[\begin{array}{r}1 \\ -1 \\ 2\end{array}\right]$.

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Question:
Consider the matrix
$$
A=\left[\begin{array}{ccc}
1 & 0 & 0 \\
0 & \cos t & -\sin t \\
0 & \sin t & \cos t
\end{array}\right]
$$
Does there exist a value of t for which this matrix fails to be invertible? Explain.

Answer: No. It has non-zero determinant $\operatorname{det}(A)=\cos ^{2} t+\sin ^{2} t=1$ for all $t$, so it is invertible for all $t$.

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Question:
Find the eigenvalues and eigenvectors of the matrix
$$
\left[\begin{array}{rrr}
8 & 0 & 10 \\
-6 & -3 & -6 \\
-5 & 0 & -7
\end{array}\right] \text {. }
$$
One eigenvalue is -3. Diagonalize if possible.

Answer:
The eigenvectors and eigenvalues are:

$$
\left\{\left[\begin{array}{l}
0 \\
1 \\
0
\end{array}\right]\right\} \text { for eigenvalue }-3, \quad\left\{\left[\begin{array}{c}
-2 \\
1 \\
1
\end{array}\right]\right\} \text { for eigenvalue } 3, \quad\left\{\left[\begin{array}{c}
-1 \\
0 \\
1
\end{array}\right]\right\} \text { for eigenvalue }-2
$$

The matrix $P$ needed to diagonalize the above matrix is

$$
\left[\begin{array}{rrr}
0 & -2 & -1 \\
1 & 1 & 0 \\
0 & 1 & 1
\end{array}\right]
$$

and the diagonal matrix $D$ is

$$
\left[\begin{array}{rrr}
-3 & 0 & 0 \\
0 & 3 & 0 \\
0 & 0 & -2
\end{array}\right]
$$

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Question: Find $2 \times 2$-matrices $A$ and $B$ such that $A \neq 0$ and $B \neq 0$, but $A B=0$.

Answer: $A=\left[\begin{array}{rr}1 & -1 \\ -1 & 1\end{array}\right], B=\left[\begin{array}{ll}1 & 1 \\ 1 & 1\end{array}\right]$

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Question: Find the matrix for $T(\mathbf{w})=\operatorname{proj}_{\mathbf{v}}(\mathbf{w})$, where $\mathbf{v}=[1,5,3]^{T}$.

Answer:
$$
\frac{1}{35}\left[\begin{array}{rrr}
1 & 5 & 3 \\
5 & 25 & 15 \\
3 & 15 & 9
\end{array}\right]
$$

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Question: Suppose there is a unique solution to a system of linear equations. What must be true of the pivot columns in the augmented matrix?

Answer: The last column must not be a pivot column. The remaining columns must each be pivot columns.

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Question:
Which of the following matrices is right invertible? Find a right inverse if one exists. If possible, find two different right inverses.

$$
A=\left[\begin{array}{lll}
1 & 2 & 3 \\
0 & 1 & 0
\end{array}\right] \quad B=\left[\begin{array}{lll}
1 & 1 & 2 \\
2 & 2 & 4
\end{array}\right] \quad C=\left[\begin{array}{ll}
1 & 2 \\
1 & 0 \\
0 & 1
\end{array}\right] \quad D=\left[\begin{array}{ll}
1 & 2 \\
2 & 3
\end{array}\right]
$$

Answer:
The matrix $A$ is right invertible. Two possible right inverses are

$$
\left[\begin{array}{cc}
1 & -2 \\
0 & 1 \\
0 & 0
\end{array}\right] \text { and }\left[\begin{array}{cc}
-2 & 1 \\
0 & 1 \\
1 & -1
\end{array}\right]
$$

The matrices $B$ and $C$ are not right invertible. The matrix $D$ is right invertible with inverse

$$
\left[\begin{array}{cc}
-3 & 2 \\
2 & -1
\end{array}\right]
$$

Since $D$ is square, its right inverse is actually an inverse, and therefore unique.

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Question:
Consider $\mathbb{R}^{4}$ with the usual dot product. Let
$$
\mathbf{u}_{1}=\left[\begin{array}{l}
0 \\
0 \\
1 \\
3
\end{array}\right], \quad \mathbf{u}_{2}=\left[\begin{array}{l}
1 \\
1 \\
0 \\
0
\end{array}\right], \quad \mathbf{u}_{3}=\left[\begin{array}{r}
1 \\
-1 \\
3 \\
-1
\end{array}\right], \quad \text { and } \quad \mathbf{v}=\left[\begin{array}{r}
6 \\
-2 \\
-5 \\
5
\end{array}\right]
$$
Note that $\mathbf{u}_{1}, \mathbf{u}_{2}$, and $\mathbf{u}_{3}$ are orthogonal. Find the best approximation of $\mathbf{v}$ in $\operatorname{span}\left\{\mathbf{u}_{1}, \mathbf{u}_{2}, \mathbf{u}_{3}\right\}$.

Answer: The best approximation is $\mathbf{v}^{\prime}=\mathbf{u}_{1}+2 \mathbf{u}_{2}-\mathbf{u}_{3}=\left[\begin{array}{r}1 \\ 3 \\ -2 \\ 4\end{array}\right]$.

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Question:
Consider the matrix
$$
A=\left[\begin{array}{ccc}
e^{t} & e^{-t} \cos t & e^{-t} \sin t \\
e^{t} & -e^{-t} \cos t-e^{-t} \sin t & -e^{-t} \sin t+e^{-t} \cos t \\
e^{t} & 2 e^{-t} \sin t & -2 e^{-t} \cos t
\end{array}\right]
$$
Does there exist a value of t for which this matrix fails to be invertible? Explain.

Answer:
$$
\operatorname{det}\left[\begin{array}{ccc}
e^{t} & e^{-t} \cos t & e^{-t} \sin t \\
e^{t} & -e^{-t} \cos t-e^{-t} \sin t & -e^{-t} \sin t+e^{-t} \cos t \\
e^{t} & 2 e^{-t} \sin t & -2 e^{-t} \cos t
\end{array}\right]=5 e^{-t} \neq 0
$$
and so this matrix is always invertible.

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Question: Consider the system $-5 x+2 y-z=0$ and $-5 x-2 y-z=0$. Both equations equal zero and so $-5 x+2 y-z=-5 x-2 y-z$ which is equivalent to $y=0$. Does it follow that $x$ and $z$ can equal anything? Notice that when $x=1, z=-4$, and $y=0$ are plugged in to the equations, the equations do not equal 0 . Why?

Answer: These are not legitimate row operations. They do not preserve the solution set of the system.

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Question:
Use elementary operations to solve the system of equations
$$
\begin{aligned}
& 3 x+y=3 \\
& x+2 y=1
\end{aligned}
$$

Answer: $(x, y)=(1,0)$.

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Question: Consider the set of all vectors $\left[\begin{array}{l}x \\ y\end{array}\right] \in \mathbb{R}^{2}$ such that $x+y \geq 0$. Is this a subspace of $\mathbb{R}^{2}$ ?

Answer: No. It is not closed under scalar multiplication.

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Question:
Find the inverse, if it exists, of the matrix
$$
A=\left[\begin{array}{ccc}
e^{t} & \cos t & \sin t \\
e^{t} & -\sin t & \cos t \\
e^{t} & -\cos t & -\sin t
\end{array}\right]
$$

Answer:
$$
\left[\begin{array}{ccc}
e^{t} & \cos t & \sin t \\
e^{t} & -\sin t & \cos t \\
e^{t} & -\cos t & -\sin t
\end{array}\right]^{-1}=\left[\begin{array}{ccc}
\frac{1}{2} e^{-t} & 0 & \frac{1}{2} e^{-t} \\
\frac{1}{2} \cos t+\frac{1}{2} \sin t & -\sin t & \frac{1}{2} \sin t-\frac{1}{2} \cos t \\
\frac{1}{2} \sin t-\frac{1}{2} \cos t & \cos t & -\frac{1}{2} \cos t-\frac{1}{2} \sin t
\end{array}\right]
$$

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Question:
Normalize the following vectors.
$$
\mathbf{u}=\left[\begin{array}{l}
1 \\
2
\end{array}\right], \quad \mathbf{v}=\left[\begin{array}{r}
-2 \\
3 \\
2
\end{array}\right], \quad \mathbf{w}=\left[\begin{array}{r}
5 \\
-3 \\
1 \\
-1
\end{array}\right]
$$

Answer:
$$
\frac{1}{\|\mathbf{u}\|} \mathbf{u}=\frac{1}{\sqrt{5}}\left[\begin{array}{l}
1 \\
2
\end{array}\right], \quad \frac{1}{\|\mathbf{v}\|} \mathbf{v}=\frac{1}{\sqrt{17}}\left[\begin{array}{r}
-2 \\
3 \\
2
\end{array}\right], \quad \frac{1}{\|\mathbf{w}\|} \mathbf{w}=\frac{1}{6}\left[\begin{array}{r}
5 \\
-3 \\
1 \\
-1
\end{array}\right]
$$

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Question: Let $X=\{1,2, \ldots, n\}$, and consider the space Func $_{X, \mathbb{R}}$ of real-valued functions defined on $X$. Explain how Func $_{X, \mathbb{R}}$ can be considered as $\mathbb{R}^{n}$.

Answer: Let $f(i)$ be the $i^{\text {th }}$ component of a vector $\mathbf{x} \in \mathbb{R}^{n}$. Thus a typical element in $\mathbb{R}^{n}$ is $(f(1), \ldots, f(n))$.

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Question:
Decide whether
$$
\mathbf{v}=\left[\begin{array}{l}
4 \\
4 \\
4
\end{array}\right]
$$
is a linear combination of the vectors
$$
\mathbf{u}_{1}=\left[\begin{array}{r}
3 \\
1 \\
-1
\end{array}\right], \quad \mathbf{u}_{2}=\left[\begin{array}{r}
8 \\
0 \\
-1
\end{array}\right] \quad \text { and } \quad \mathbf{u}_{3}=\left[\begin{array}{r}
2 \\
-2 \\
1
\end{array}\right]
$$
If yes, find the coefficients.

Answer:
The system

$$
\left[\begin{array}{l}
4 \\
4 \\
4
\end{array}\right]=a_{1}\left[\begin{array}{r}
3 \\
1 \\
-1
\end{array}\right]+a_{2}\left[\begin{array}{r}
8 \\
0 \\
-1
\end{array}\right]+a_{3}\left[\begin{array}{r}
2 \\
-2 \\
1
\end{array}\right]
$$

has no solution.

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Question: Is it possible for a non-zero matrix to have only 0 as an eigenvalue?

Answer: Yes. $\left[\begin{array}{ll}0 & 1 \\ 0 & 0\end{array}\right]$ works.

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Question: Let $X=\left[\begin{array}{lll}-1 & -1 & 1\end{array}\right]$ and $Y=\left[\begin{array}{lll}0 & 1 & 2\end{array}\right]$. Find $X^{T} Y$ and $X Y^{T}$ if possible.

Answer: $X^{T} Y=\left[\begin{array}{rrr}0 & -1 & -2 \\ 0 & -1 & -2 \\ 0 & 1 & 2\end{array}\right], X Y^{T}=1$

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Question: Find $2 \times 2$-matrices $A, B$, and $C$ such that $A \neq 0, C \neq B$, but $A C=A B$.

Answer: $A=\left[\begin{array}{rr}1 & -1 \\ -1 & 1\end{array}\right], B=\left[\begin{array}{ll}1 & 1 \\ 1 & 1\end{array}\right], C=\left[\begin{array}{ll}2 & 2 \\ 2 & 2\end{array}\right]$

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Question:
Encrypt the message "Rendezvous at dawn" using the Hill cipher with block size 3 and encryption matrix
$$
A=\left[\begin{array}{lll}
2 & 1 & 1 \\
1 & 3 & 1 \\
1 & 1 & 4
\end{array}\right]
$$

Answer: Ciphertext: "ZRUJPZVAEJTWOXGJZV".

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Question:
Solve the following two systems of equations simultaneously, by using a single augmented matrix with two constant vectors.
$$
\begin{array}{rrr}
x+2 y-z=0 & x+2 y-z=1 \\
2 x+3 y+z=3 & 2 x+3 y+z=7 \\
x-y+2 z=3 & x-y+2 z=4
\end{array}
$$

Answer: The rank of the coefficient matrix is 3 , so both systems have a unique solution. The solution of the first system is $(x, y, z)=(1,0,1)$, and the solution of the second system is $(x, y, z)=(1,1,2)$.

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Question: Suppose $\mathbf{u}, \mathbf{v}$, and $\mathbf{w}$ are three vectors whose components are all integers. Can you conclude the volume of the parallelepiped determined from these three vectors will always be an integer?

Answer: Yes. It will involve the sum of a product of integers and so it will be an integer.

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Question:
Consider the matrix
$$
A=\left[\begin{array}{rrr}
1 & t & t^{2} \\
0 & 1 & 2 t \\
t & 0 & 2
\end{array}\right]
$$
Does there exist a value of t for which this matrix fails to be invertible? Explain.

Answer: $\operatorname{det}(A)=\left|\begin{array}{ccc}1 & t & t^{2} \\ 0 & 1 & 2 t \\ t & 0 & 2\end{array}\right|=t^{3}+2$, and so $A$ has no inverse when $t=-\sqrt[3]{2}$.

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Question:
Find scalars $x, y, z$ such that the following two matrices are equal.
$$
\left[\begin{array}{rr}
x & -1 \\
2 & 4
\end{array}\right] \text { and }\left[\begin{array}{ll}
2 & y \\
z & 4
\end{array}\right]
$$

Answer: $x=2, y=-1, z=2$.

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Question: Find the area of the parallelogram determined by the vectors $\left[\begin{array}{l}1 \\ 2 \\ 3\end{array}\right],\left[\begin{array}{r}3 \\ -2 \\ 1\end{array}\right]$.

Answer: $\left[\begin{array}{l}1 \\ 1 \\ 3\end{array}\right] \times\left[\begin{array}{r}-7 \\ -2 \\ 2\end{array}\right]=\left[\begin{array}{r}8 \\ -23 \\ 5\end{array}\right]$. The area of the parallelogram is $8 \sqrt{3}$.

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Question:
Let $\mathbf{v}=\left[\begin{array}{r}1 \\ -2 \\ 3\end{array}\right]$ and consider the linear function $T(\mathbf{w})=\operatorname{proj}_{\mathbf{v}}(\mathbf{w})$. Find the matrix of $T$ with respect to the basis

$$
B=\left\{\mathbf{v}_{1}, \mathbf{v}_{2}, \mathbf{v}_{3}\right\}=\left\{\left[\begin{array}{r}
1 \\
-2 \\
3
\end{array}\right],\left[\begin{array}{l}
2 \\
1 \\
0
\end{array}\right],\left[\begin{array}{l}
3 \\
0 \\
1
\end{array}\right]\right\}
$$

Answer:
We have $T\left(\mathbf{v}_{1}\right)=\mathbf{v}_{1}, T\left(\mathbf{v}_{2}\right)=\mathbf{0}$, and $T\left(\mathbf{v}_{3}\right)=\mathbf{0}$. Therefore

$$
[T]_{B, B}=\left[\begin{array}{ccc}
1 & 0 & 0 \\
0 & 0 & 0 \\
0 & 0 & 0
\end{array}\right] .
$$

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Question: Let $B=\left\{\left[\begin{array}{r}1 \\ -1 \\ 2\end{array}\right],\left[\begin{array}{l}2 \\ 1 \\ 2\end{array}\right],\left[\begin{array}{r}-1 \\ 0 \\ 2\end{array}\right]\right\}$ be a basis of $\mathbb{R}^{3}$ and let $\mathbf{x}=\left[\begin{array}{r}5 \\ -1 \\ 4\end{array}\right]$ be a vector in $\mathbb{R}^{2}$. Find $[\mathbf{x}]_{B}$.

Answer: $[\mathbf{x}]_{B}=\left[\begin{array}{r}2 \\ 1 \\ -1\end{array}\right]$.

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Question:
Find the value of $y$ in the following system of equations:
$$
\left[\begin{array}{ccc}
1 & 1 & 1 \\
1 & t & t^{2} \\
1 & s & s^{2}
\end{array}\right]\left[\begin{array}{l}
x \\
y \\
z
\end{array}\right]=\left[\begin{array}{l}
t \\
s \\
1
\end{array}\right] .
$$

Answer:
By Cramer's rule, we have
$$
y=\frac{\left|\begin{array}{ccc}
1 & t & 1 \\
1 & s & t^{2} \\
1 & 1 & s^{2}
\end{array}\right|}{\left|\begin{array}{ccc}
1 & 1 & 1 \\
1 & t & t^{2} \\
1 & s & s^{2}
\end{array}\right|}=\frac{s^{3}+t^{3}+1-2-t s^{2}-t^{2}}{t s^{2}+t^{2}+s-t-s t^{2}-s^{2}} .
$$

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Question: Let $U=\left\{[x, y, z]^{T} \in \mathbb{R}^{3}|| x \mid \leq 4\right\}$. Is $U$ a subspace of $\mathbb{R}^{3}$ ?

Answer: This is not a subspace. $\left[\begin{array}{l}1 \\ 1 \\ 1\end{array}\right]$ is in it, but $5\left[\begin{array}{l}1 \\ 1 \\ 1\end{array}\right]$ is not.

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Question:
Consider the following augmented matrix in which $*$ denotes an arbitrary number and denotes a non-zero number. Determine whether the given augmented matrix is consistent. If consistent, is the solution unique?
$$
\left[\begin{array}{ccc|c}
\mathbf{\square} & * & * & * \\
0 & \mathbf{0} & * & * \\
0 & 0 & \mathbf{\square} & *
\end{array}\right]
$$

Answer: A solution exists and is unique.

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Question: Find the distance between the points $P=(1,3,-1,0)$ and $Q=(2,2,3,3)$ in $\mathbb{R}^{4}$.

Answer: $\sqrt{27}$.

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Question:
Consider the following vector equation for a plane in $\mathbb{R}^{4}$ :
$$
\left[\begin{array}{l}
x \\
y \\
z \\
w
\end{array}\right]=\left[\begin{array}{l}
1 \\
2 \\
0 \\
0
\end{array}\right]+t\left[\begin{array}{l}
1 \\
0 \\
0 \\
1
\end{array}\right]+s\left[\begin{array}{r}
-1 \\
-1 \\
1 \\
0
\end{array}\right] .
$$
Find a new vector equation for the same plane by doing the change of parameters $t=1-r_{1}, s=r_{1}+r_{2}$.

Answer:
We have

$$
\left[\begin{array}{l}
x \\
y \\
z \\
w
\end{array}\right]=\left[\begin{array}{l}
1 \\
2 \\
0 \\
0
\end{array}\right]+\left(1-r_{1}\right)\left[\begin{array}{l}
1 \\
0 \\
0 \\
1
\end{array}\right]+\left(r_{1}+r_{2}\right)\left[\begin{array}{r}
-1 \\
-1 \\
1 \\
0
\end{array}\right]=\left[\begin{array}{l}
2 \\
2 \\
0 \\
1
\end{array}\right]+r_{1}\left[\begin{array}{r}
-2 \\
-1 \\
1 \\
-1
\end{array}\right]+r_{2}\left[\begin{array}{r}
-1 \\
-1 \\
1 \\
0
\end{array}\right] .
$$

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Question: Solve the system of equations $7 x+14 y+15 z=22,2 x+4 y+3 z=5$, and $3 x+6 y+10 z=$ 13.

Answer: Solution is: $[x=1-2 t, z=1, y=t]$.

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Question: Is $\mathbf{u} \times(\mathbf{v} \times \mathbf{w})=(\mathbf{u} \times \mathbf{v}) \times \mathbf{w}$ ? What is the meaning of $\mathbf{u} \times \mathbf{v} \times \mathbf{w}$ ? Explain. Hint: Try $(\mathbf{i} \times \mathbf{j}) \times \mathbf{j}$.

Answer: $(\mathbf{i} \times \mathbf{j}) \times \mathbf{j}=\mathbf{k} \times \mathbf{j}=-\mathbf{i}$. However, $\mathbf{i} \times(\mathbf{j} \times \mathbf{j})=\mathbf{0}$ and so the cross product is not associative. The expression $\mathbf{u} \times \mathbf{v} \times \mathbf{w}$ has no meaning.

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Question: Let $A=\left[\begin{array}{rr}-5 & -6 \\ 9 & 10\end{array}\right]$. Find a square root of $A$, i.e., find a matrix $B$ such that $B^{2}=A$.

Answer: $B=\left[\begin{array}{rr}-1 & -2 \\ 3 & 4\end{array}\right]$.

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Question: Suppose $A B=A C$ and $A$ is an invertible $n \times n$-matrix. Does it follow that $B=C$ ? Explain why or why not.

Answer: Yes $B=C$. Multiply $A B=A C$ on the left by $A^{-1}$.

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Question: Solve the system of equations $9 x-2 y+4 z=-17,13 x-3 y+6 z=-25$, and $-2 x-z=3$.

Answer: Solution is: $[x=-1, y=2, z=-1]$.

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Question: Consider the set of all vectors $\left[\begin{array}{l}x \\ y\end{array}\right] \in \mathbb{R}^{2}$ such that $x y=0$. Is this a subspace of $\mathbb{R}^{2}$ ?

Answer: No. It is not closed under addition.

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Question: Suppose a system of linear equations has an augmented matrix with 2 rows and 4 columns and the last column is a pivot column. Could the system of linear equations be consistent? Explain.

Answer: No. This would lead to $0=1$.

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Question: Let $\mathbf{v}=\left[\begin{array}{r}1 \\ -2 \\ 3\end{array}\right]$ and consider the linear function $T(\mathbf{w})=\operatorname{proj}_{\mathbf{v}}(\mathbf{w})$. Find the matrix of $T$ with respect to the standard basis of $\mathbb{R}^{3}$.

Answer:
Recall that $\operatorname{proj}_{\mathbf{v}}(\mathbf{w})=\frac{\mathbf{v}^{\bullet} \mathbf{w}}{\|\mathbf{v}\|^{2}} \mathbf{v}$. The desired matrix has $i^{\text {th }}$ column equal to $\operatorname{proj}_{\mathbf{v}}\left(\mathbf{e}_{i}\right)$. Therefore, the desired matrix is

$$
\frac{1}{14}\left[\begin{array}{rrr}
1 & -2 & 3 \\
-2 & 4 & -6 \\
3 & -6 & 9
\end{array}\right] .
$$

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Question: Consider $\mathbb{C}^{3}$ with the complex dot product. Let $\mathbf{v}_{1}=\left[\begin{array}{c}0 \\ i \\ 2\end{array}\right]$ and $\mathbf{v}_{2}=\left[\begin{array}{c}1 \\ 1+i \\ 3 i+2\end{array}\right]$. Use the Gram-Schmidt procedure to find an orthogonal basis for $\operatorname{span}\left\{\mathbf{v}_{1}, \mathbf{v}_{2}\right\}$. Then find an orthonormal basis.

Answer: Orthogonal: $[0, i, 2]^{T},[1,2, i]^{T}$. Orthonormal: $\frac{1}{\sqrt{5}}[0, i, 2]^{T}, \frac{1}{\sqrt{6}}[1,2, i]^{T}$.

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Question:
Suppose $B=\left\{\mathbf{u}_{1}, \mathbf{u}_{2}, \mathbf{u}_{3}\right\}$ is an orthogonal basis of $\mathbb{R}^{3}$. We have been told that
$$
\mathbf{u}_{1}=\left[\begin{array}{l}
1 \\
1 \\
0
\end{array}\right],
$$
but it is not known what $\mathbf{u}_{2}$ and $\mathbf{u}_{3}$ are. Find the first coordinate of the vector
$$
\mathbf{v}=\left[\begin{array}{l}
1 \\
0 \\
2
\end{array}\right]
$$
with respect to the basis $B$.

Answer: We have $\mathbf{v}=a_{1} \mathbf{u}_{1}+a_{2} \mathbf{u}_{2}+a_{3} \mathbf{u}_{3}$ where $a_{1}=\frac{\left\langle\mathbf{u}_{1}, \mathbf{v}\right\rangle}{\left\langle\mathbf{u}_{1}, \mathbf{u}_{1}\right\rangle}=\frac{1}{2}$. So the first coordinate is $\frac{1}{2}$.

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Question: Describe the span of the vectors $\mathbf{u}=\left[\begin{array}{l}1 \\ 0 \\ 2\end{array}\right]$ and $\mathbf{v}=\left[\begin{array}{r}2 \\ -1 \\ 1\end{array}\right]$ in $\mathbb{R}^{3}$.

Answer: It is the plane $2 x+3 y-z=0$.

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Question:
Consider the following augmented matrix in which $*$ denotes an arbitrary number and denotes a non-zero number. Determine whether the given augmented matrix is consistent. If consistent, is the solution unique?
$$
\left[\begin{array}{lllll|l}
\mathbf{\square} & * & * & * & * & * \\
0 & \mathbf{\square} & * & * & 0 & * \\
0 & 0 & \mathbf{\square} & * & * & * \\
0 & 0 & 0 & 0 & \mathbf{\square} & *
\end{array}\right]
$$

Answer: The solution exists but is not unique.

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Question: Solve the system of equations $65 x+84 y+16 z=546,81 x+105 y+20 z=682$, and $84 x+110 y+21 z=713$.

Answer: Solution is: $[x=2, y=4, z=5]$.

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Question: Let $A=\left[\begin{array}{ll}1 & 2 \\ 3 & 4\end{array}\right]$ and $B=\left[\begin{array}{ll}1 & 2 \\ 1 & k\end{array}\right]$. Is it possible to choose $k$ such that $A B=B A$? If so, what should $k$ equal?

Answer: There is no possible choice of $k$ which will make these matrices commute.

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Question: Suppose there is a unique solution to a system of linear equations. What must be true of the pivot columns in the augmented matrix?

Answer: The last column must not be a pivot column. The remaining columns must each be pivot columns.


Question: An object moves 10 meters in the direction of $\vec{j}+\vec{i}$. There are two forces acting on this object, $\vec{F}_{1}=\vec{i}+2 \vec{j}+2 \vec{k}$, and $\vec{F}_{2}=5 \vec{i}+2 \vec{j}-6 \vec{k}$. Find the total work done on the object by the two forces. Hint: You can take the work done by the resultant of the two forces or you can add the work done by each force. Why?

Answer:
$$
\begin{aligned}
\vec{F}_{1} \bullet\left[\begin{array}{c}
\frac{1}{\sqrt{2}} \\
\frac{1}{\sqrt{2}} \\
0
\end{array}\right] 10+\vec{F}_{2} \bullet\left[\begin{array}{c}
\frac{1}{\sqrt{2}} \\
\frac{1}{\sqrt{2}} \\
0
\end{array}\right] 10 & =\left(\vec{F}_{1}+\vec{F}_{2}\right) \bullet\left[\begin{array}{c}
\frac{1}{\sqrt{2}} \\
\frac{1}{\sqrt{2}} \\
0
\end{array}\right] 10 \\
& =\left[\begin{array}{r}
6 \\
4 \\
-4
\end{array}\right] \bullet\left[\begin{array}{c}
\frac{1}{\sqrt{2}} \\
\frac{1}{\sqrt{2}} \\
0
\end{array}\right] 10 \\
& =50 \sqrt{2}
\end{aligned}
$$


Question: Find the matrix for $T(\vec{w})=\operatorname{proj}_{\vec{v}}(\vec{w})$ where $\vec{v}=\left[\begin{array}{lll}1 & 0 & 3\end{array}\right]^{T}$.

Answer:
$$
\frac{1}{10}\left[\begin{array}{lll}
1 & 0 & 3 \\
0 & 0 & 0 \\
3 & 0 & 9
\end{array}\right]
$$


Question:
Determine if the system is consistent. If so, is the solution unique?
$$
\begin{gathered}
x+2 y+z-w=2 \\
x-y+z+w=0 \\
2 x+y-z=1 \\
4 x+2 y+z=3
\end{gathered}
$$

Answer: Solution is: $\left[w=\frac{3}{2} y-1, x=\frac{2}{3}-\frac{1}{2} y, z=\frac{1}{3}\right]$


Question: Let $M=\left\{\vec{u}=\left(u_{1}, u_{2}, u_{3}, u_{4}\right) \in \mathbb{R}^{4}:\left|u_{1}\right| \leq 4\right\}$. Is $M$ a subspace of $\mathbb{R}^{4}$ ?

Answer: This is not a subspace. $\left[\begin{array}{l}1 \\ 1 \\ 1 \\ 1\end{array}\right]$ is in it, but $20\left[\begin{array}{l}1 \\ 1 \\ 1 \\ 1\end{array}\right]$ is not.


Question: Find the solution to the system of equations, $-8 x+2 y+5 z=18,-8 x+3 y+5 z=13$, and $-4 x+y+5 z=19$.

Answer: Solution is: $[x=-1, y=-5, z=4]$


Question: Find the area of the triangle determined by the three points, $(1,2,3),(4,2,0)$ and $(-3,2,1)$.

Answer: $\left[\begin{array}{r}3 \\ 0 \\ -3\end{array}\right] \times\left[\begin{array}{r}-4 \\ 0 \\ -2\end{array}\right]=\left[\begin{array}{r}0 \\ 18 \\ 0\end{array}\right]$. So the area is 9.


Question:
Write the solution set of the following system as a linear combination of vectors
$$
\left[\begin{array}{rrrr}
1 & 1 & 0 & 1 \\
1 & -1 & 1 & 0 \\
3 & 1 & 1 & 2 \\
3 & 3 & 0 & 3
\end{array}\right]\left[\begin{array}{l}
x \\
y \\
z \\
w
\end{array}\right]=\left[\begin{array}{l}
0 \\
0 \\
0 \\
0
\end{array}\right]
$$

Answer:
Solution is:

$$
\left[\begin{array}{c}
-\frac{1}{2} s-\frac{1}{2} t \\
\frac{1}{2} s-\frac{1}{2} t \\
s \\
t
\end{array}\right]
$$

for $s, t \in \mathbb{R}$. A basis is

$$
\left\{\left[\begin{array}{r}
-1 \\
1 \\
2 \\
0
\end{array}\right],\left[\begin{array}{c}
-1 \\
1 \\
0 \\
1
\end{array}\right]\right\}
$$


Question: Find the matrix for $T(\vec{w})=\operatorname{proj}_{\vec{v}}(\vec{w})$ where $\vec{v}=\left[\begin{array}{lll}1 & -2 & 3\end{array}\right]^{T}$.

Answer:
Recall that $\operatorname{proj}_{\vec{u}}(\vec{v})=\frac{\vec{v} \vec{u}}{\|\vec{u}\|^{2}} \vec{u}$ and so the desired matrix has $i^{t h}$ column equal to $\operatorname{proj}_{\vec{u}}\left(\vec{e}_{i}\right)$. Therefore, the matrix desired is

$$
\frac{1}{14}\left[\begin{array}{rrr}
1 & -2 & 3 \\
-2 & 4 & -6 \\
3 & -6 & 9
\end{array}\right]
$$


Question: Is it possible for a nonzero matrix to have only 0 as an eigenvalue?

Answer: Yes. $\left[\begin{array}{ll}0 & 1 \\ 0 & 0\end{array}\right]$ works.


Question: Four times the weight of Gaston is 150 pounds more than the weight of Ichabod. Four times the weight of Ichabod is 660 pounds less than seventeen times the weight of Gaston. Four times the weight of Gaston plus the weight of Siegfried equals 290 pounds. Brunhilde would balance all three of the others. Find the weights of the four people.

Answer: Solution is : $\{g=60, I=90, b=200, s=50\}$


Question: Find the matrix for $T(\vec{w})=\operatorname{proj}_{\vec{v}}(\vec{w})$ where $\vec{v}=\left[\begin{array}{lll}1 & -2 & 3\end{array}\right]^{T}$.

Answer:
Recall that $\operatorname{proj}_{\vec{u}}(\vec{v})=\frac{\vec{v} \bullet \vec{u}}{\|\vec{u}\|^{2}} \vec{u}$ and so the desired matrix has $i^{t h}$ column equal to $\operatorname{proj}_{\vec{u}}\left(\vec{e}_{i}\right)$. Therefore, the matrix desired is

$$
\frac{1}{14}\left[\begin{array}{rrr}
1 & -2 & 3 \\
-2 & 4 & -6 \\
3 & -6 & 9
\end{array}\right]
$$


Question:
Write the solution set of the following system as a linear combination of vectors
$$
\left[\begin{array}{rrrr}
1 & 1 & 0 & 1 \\
2 & 1 & 1 & 2 \\
1 & 0 & 1 & 1 \\
0 & -1 & 1 & 1
\end{array}\right]\left[\begin{array}{l}
x \\
y \\
z \\
w
\end{array}\right]=\left[\begin{array}{l}
0 \\
0 \\
0 \\
0
\end{array}\right]
$$

Answer: Solution is: $\left[\begin{array}{r}-\hat{t} \\ \hat{t} \\ \hat{t} \\ 0\end{array}\right]$, a basis is $\left[\begin{array}{l}1 \\ 1 \\ 1 \\ 0\end{array}\right]$.


Question: Find the complex cube roots of 8.

Answer: Solution is: $i \sqrt{3}+1,1-i \sqrt{3},-2$


Question: Decide whether $\vec{v}=\left[\begin{array}{r}4 \\ 4 \\ -3\end{array}\right]$ is a linear combination of the vectors $\vec{u}_{1}=\left[\begin{array}{r}3 \\ 1 \\ -1\end{array}\right]$ and $\vec{u}_{2}=\left[\begin{array}{r}2 \\ -2 \\ 1\end{array}\right]$.

Answer:
$$
\left[\begin{array}{r}
4 \\
4 \\
-3
\end{array}\right]=2\left[\begin{array}{r}
3 \\
1 \\
-1
\end{array}\right]-\left[\begin{array}{r}
2 \\
-2 \\
1
\end{array}\right]
$$


Question:
Write the solution set of the following system as a linear combination of vectors.
$$
\left[\begin{array}{lll}
1 & -1 & 2 \\
1 & -2 & 0 \\
3 & -4 & 4
\end{array}\right]\left[\begin{array}{l}
x \\
y \\
z
\end{array}\right]=\left[\begin{array}{l}
0 \\
0 \\
0
\end{array}\right]
$$

Answer: Solution is: $\left[\begin{array}{r}-4 \hat{t} \\ -2 \hat{t} \\ \hat{t}\end{array}\right]$. A basis is $\left[\begin{array}{r}-4 \\ -2 \\ 1\end{array}\right]$


Question: Let $A=\left[\begin{array}{ll}1 & 2 \\ 2 & 1\end{array}\right]$. Diagonalize A to find $A^{10}$.

Answer:
First we write $A=P D P^{-1}$.

$$
\left[\begin{array}{ll}
1 & 2 \\
2 & 1
\end{array}\right]=\left[\begin{array}{rr}
-1 & 1 \\
1 & 1
\end{array}\right]\left[\begin{array}{rr}
-1 & 0 \\
0 & 3
\end{array}\right]\left[\begin{array}{rr}
-\frac{1}{2} & \frac{1}{2} \\
\frac{1}{2} & \frac{1}{2}
\end{array}\right]
$$

Therefore $A^{10}=P D^{10} P^{-1}$.

$$
\begin{aligned}
{\left[\begin{array}{ll}
1 & 2 \\
2 & 1
\end{array}\right]^{10} } & =\left[\begin{array}{rr}
-1 & 1 \\
1 & 1
\end{array}\right]\left[\begin{array}{rr}
-1 & 0 \\
0 & 3^{10}
\end{array}\right]\left[\begin{array}{rr}
-\frac{1}{2} & \frac{1}{2} \\
\frac{1}{2} & \frac{1}{2}
\end{array}\right] \\
& =\left[\begin{array}{ll}
29525 & 29524 \\
29524 & 29525
\end{array}\right]
\end{aligned}
$$


Question:
Let
$$
A=\left[\begin{array}{ll}
0 & 1 \\
5 & 3
\end{array}\right]
$$
Find $A^{-1}$ if possible. If $A^{-1}$ does not exist, explain why.

Answer: $\left[\begin{array}{ll}0 & 1 \\ 5 & 3\end{array}\right]^{-1}=\left[\begin{array}{cc}-\frac{3}{5} & \frac{1}{5} \\ 1 & 0\end{array}\right]$


Question: Find the solution to the system of equations, $-9 x+15 y=66,-11 x+18 y=79,-x+y=4$, and $z=3$.

Answer: Solution is: $[x=1, y=5, z=3]$


Question:
Consider the matrix
$$
A=\left[\begin{array}{ccc}
e^{t} & \cosh t & \sinh t \\
e^{t} & \sinh t & \cosh t \\
e^{t} & \cosh t & \sinh t
\end{array}\right]
$$
Does there exist a value of $t$ for which this matrix fails to have an inverse? Explain.

Answer:
$$
\operatorname{det}\left[\begin{array}{ccc}
e^{t} & \cosh t & \sinh t \\
e^{t} & \sinh t & \cosh t \\
e^{t} & \cosh t & \sinh t
\end{array}\right]=0
$$

and so this matrix fails to have a nonzero determinant at any value of $t$.


Question: Suppose a system of linear equations has a $2 \times 4$ augmented matrix and the last column is a pivot column. Could the system of linear equations be consistent? Explain.

Answer: No. This would lead to $0=1$.


Question: Suppose the characteristic polynomial of an $n \times n$ matrix $A$ is $1-X^{n}$. Find $A^{m n}$ where $m$ is an integer.

Answer:
The eigenvalues are distinct because they are the $n^{\text {th }}$ roots of 1 . Hence if $X$ is a given vector with

$$
X=\sum_{j=1}^{n} a_{j} V_{j}
$$

then

$$
A^{n m} X=A^{n m} \sum_{j=1}^{n} a_{j} V_{j}=\sum_{j=1}^{n} a_{j} A^{n m} V_{j}=\sum_{j=1}^{n} a_{j} V_{j}=X
$$

so $A^{n m}=I$.


Question:
Find $h$ such that
$$
\left[\begin{array}{rr|r}
1 & 1 & 4 \\
3 & h & 12
\end{array}\right]
$$
is the augmented matrix of a consistent system.

Answer: Any $h$ will work.


Question:
Find the following determinant by expanding along the first row and second column.

$$
\left|\begin{array}{lll}
1 & 2 & 1 \\
2 & 1 & 3 \\
2 & 1 & 1
\end{array}\right|
$$

Answer:
$$
\left|\begin{array}{lll}
1 & 2 & 1 \\
2 & 1 & 3 \\
2 & 1 & 1
\end{array}\right|=6
$$


Question:
Find the eigenvalues and eigenvectors of the matrix
$$
\left[\begin{array}{rrr}
89 & 38 & 268 \\
14 & 2 & 40 \\
-30 & -12 & -90
\end{array}\right]
$$
One eigenvalue is -3. Diagonalize if possible.

Answer:
The eigenvectors and eigenvalues are:

$$
\left\{\left[\begin{array}{l}
-6 \\
-1 \\
-2
\end{array}\right]\right\} \leftrightarrow 6,\left\{\left[\begin{array}{c}
-5 \\
-2 \\
2
\end{array}\right]\right\} \leftrightarrow-3,\left\{\left[\begin{array}{c}
-8 \\
-2 \\
3
\end{array}\right]\right\} \leftrightarrow-2
$$

The matrix $P$ needed to diagonalize the above matrix is

$$
\left[\begin{array}{rrr}
-6 & -5 & -8 \\
-1 & -2 & -2 \\
2 & 2 & 3
\end{array}\right]
$$

and the diagonal matrix $D$ is

$$
\left[\begin{array}{rrr}
6 & 0 & 0 \\
0 & -3 & 0 \\
0 & 0 & -2
\end{array}\right]
$$


Question:
Consider the matrix
$$
A=\left[\begin{array}{ccc}
e^{t} & e^{-t} \cos t & e^{-t} \sin t \\
e^{t} & -e^{-t} \cos t-e^{-t} \sin t & -e^{-t} \sin t+e^{-t} \cos t \\
e^{t} & 2 e^{-t} \sin t & -2 e^{-t} \cos t
\end{array}\right]
$$
Does there exist a value of $t$ for which this matrix fails to have an inverse? Explain.

Answer:
$$
\operatorname{det}\left[\begin{array}{ccc}
e^{t} & e^{-t} \cos t & e^{-t} \sin t \\
e^{t} & -e^{-t} \cos t-e^{-t} \sin t & -e^{-t} \sin t+e^{-t} \cos t \\
e^{t} & 2 e^{-t} \sin t & -2 e^{-t} \cos t
\end{array}\right]=5 e^{-t} \neq 0
$$

and so this matrix is always invertible.


Question:
An operation is done to get from the first matrix to the second. Identify what was done and tell how it will affect the value of the determinant.

$$
\left[\begin{array}{ll}
a & b \\
c & d
\end{array}\right] \rightarrow \cdots \rightarrow\left[\begin{array}{ll}
a & c \\
b & d
\end{array}\right]
$$

Answer: It does not change the determinant. This was just taking the transpose.


Question:
Let
$$
A=\left[\begin{array}{rr}
2 & 1 \\
-1 & 3
\end{array}\right]
$$
Find $A^{-1}$ if possible. If $A^{-1}$ does not exist, explain why.

Answer: $\left[\begin{array}{rr}2 & 1 \\ -1 & 3\end{array}\right]^{-1}=\left[\begin{array}{rr}\frac{3}{7} & -\frac{1}{7} \\ \frac{1}{7} & \frac{2}{7}\end{array}\right]$


Question:
A person sets off on a random walk with three possible locations. The Markov matrix of probabilities $A=\left[a_{i j}\right]$ is given by
$$
\left[\begin{array}{rrr}
0.5 & 0.1 & 0.6 \\
0.2 & 0.9 & 0.2 \\
0.3 & 0 & 0.2
\end{array}\right]
$$
It is unknown where the walker starts, but the probability of starting in each location is given by
$$
X_{0}=\left[\begin{array}{r}
0.2 \\
0.25 \\
0.55
\end{array}\right]
$$
What is the probability of the walker being in location 1 at time $n=2$ ?

Answer:
$$
X_{2}=\left[\begin{array}{l}
0.367 \\
0.4625 \\
0.1705
\end{array}\right]
$$

Therefore the probability of ending up in location 1 is 0.367 .


Question:
Consider the matrix
$$
A=\left[\begin{array}{rrr}
1 & t & t^{2} \\
0 & 1 & 2 t \\
t & 0 & 2
\end{array}\right]
$$
Does there exist a value of t for which this matrix fails to have an inverse? Explain.

Answer:
$$
\operatorname{det}\left[\begin{array}{ccc}
1 & t & t^{2} \\
0 & 1 & 2 t \\
t & 0 & 2
\end{array}\right]=t^{3}+2
$$

and so it has no inverse when $t=-\sqrt[3]{2}$


Question:
Let
$$
A=\left[\begin{array}{ll}
2 & 1 \\
3 & 0
\end{array}\right]
$$
Find $A^{-1}$ if possible. If $A^{-1}$ does not exist, explain why.

Answer: $\left[\begin{array}{ll}2 & 1 \\ 3 & 0\end{array}\right]^{-1}=\left[\begin{array}{cc}0 & \frac{1}{3} \\ 1 & -\frac{2}{3}\end{array}\right]$


Question: Explain why an $n \times n$ matrix $A$ is both one to one and onto if and only if its rank is $n$.

Answer: The rank is $n$ is the same as saying the columns are independent which is the same as saying $A$ is one to one which is the same as saying the columns are a basis. Thus the span of the columns of $A$ is all of $\mathbb{R}^{n}$ and so $A$ is onto. If $A$ is onto, then the columns must be linearly independent since otherwise the span of these columns would have dimension less than $n$ and so the dimension of $\mathbb{R}^{n}$ would be less than $n$.


Question: Find $\operatorname{proj}_{\vec{v}}(\vec{w})$ where $\vec{w}=\left[\begin{array}{r}1 \\ 2 \\ -2\end{array}\right]$ and $\vec{v}=\left[\begin{array}{l}1 \\ 0 \\ 3\end{array}\right]$.

Answer: $\overrightarrow{\vec{u} \bullet \vec{v}} \overrightarrow{\vec{u}} \vec{u}=\frac{-5}{10}\left[\begin{array}{l}1 \\ 0 \\ 3\end{array}\right]=\left[\begin{array}{r}-\frac{1}{2} \\ 0 \\ -\frac{3}{2}\end{array}\right]$


Question: How much work does it take to slide a crate 20 meters along a loading dock by pulling on it with a 200 Newton force at an angle of $30^{\circ}$ from the horizontal? Express your answer in Newton meters.

Answer: $200\left(\cos \left(\frac{\pi}{6}\right)\right) 20=3464.1$


Question: An airplane is flying due north at 150.0 miles per hour but it is not actually going due North because there is a wind which is pushing the airplane due east at 40.0 miles per hour. After one hour, the plane starts flying $30^{\circ}$ East of North. Assuming the plane starts at $(0,0)$, where is it after 2 hours? Let North be the direction of the positive y axis and let East be the direction of the positive $x$ axis.

Answer: After two hours it is then at $(40,150)+150\left[\begin{array}{ll}\frac{1}{2} & \frac{\sqrt{3}}{2}\end{array}\right]+\left[\begin{array}{ll}40 & 0\end{array}\right]=\left[\begin{array}{ll}155 & 75 \sqrt{3}+150\end{array}\right]=\left[\begin{array}{ll}155.0 & 279.9\end{array}\right]$


Question:
Find the eigenvalues and an orthonormal basis of eigenvectors for $A$. Diagonalize $A$ by finding an orthogonal matrix $U$ and a diagonal matrix $D$ such that $U^{T} A U=D$.
$$
A=\left[\begin{array}{rrr}
-1 & 1 & 1 \\
1 & -1 & 1 \\
1 & 1 & -1
\end{array}\right]
$$
Hint: One eigenvalue is -2.

Answer:
The eigenvectors and eigenvalues are:

$$
\left\{\left[\begin{array}{c}
\frac{1}{3} \sqrt{3} \\
\frac{1}{3} \sqrt{3} \\
\frac{1}{3} \sqrt{3}
\end{array}\right]\right\} \leftrightarrow 1,\left\{\left[\begin{array}{c}
-\frac{1}{2} \sqrt{2} \\
\frac{1}{2} \sqrt{2} \\
0
\end{array}\right],\left[\begin{array}{c}
-\frac{1}{6} \sqrt{6} \\
-\frac{1}{6} \sqrt{6} \\
\frac{1}{3} \sqrt{2} \sqrt{3}
\end{array}\right]\right\} \leftrightarrow-2 \\
{\left[\begin{array}{ccc}
\sqrt{3} / 3 & -\sqrt{2} / 2 & -\sqrt{6} / 6 \\
\sqrt{3} / 3 & \sqrt{2} / 2 & -\sqrt{6} / 6 \\
\sqrt{3} / 3 & 0 & \frac{1}{3} \sqrt{2} \sqrt{3}
\end{array}\right]^{T}\left[\begin{array}{rrr}
-1 & 1 & 1 \\
1 & -1 & 1 \\
1 & 1 & -1
\end{array}\right]} \\
.\left[\begin{array}{ccc}
\sqrt{3} / 3 & -\sqrt{2} / 2 & -\sqrt{6} / 6 \\
\sqrt{3} / 3 & \sqrt{2} / 2 & -\sqrt{6} / 6 \\
\sqrt{3} / 3 & 0 & \frac{1}{3} \sqrt{2} \sqrt{3}
\end{array}\right] \\
=\left[\begin{array}{ccc}
1 & 0 & 0 \\
0 & -2 & 0 \\
0 & 0 & -2
\end{array}\right]
$$


Question:
Find the eigenvalues and an orthonormal basis of eigenvectors for $A$. Diagonalize $A$ by finding an orthogonal matrix $U$ and a diagonal matrix $D$ such that $U^{T} A U=D$.
$$
A=\left[\begin{array}{rrr}
17 & -7 & -4 \\
-7 & 17 & -4 \\
-4 & -4 & 14
\end{array}\right]
$$
Hint: Two eigenvalues are 18 and 24.

Answer:
The eigenvectors and eigenvalues are:

$$
\left\{\left[\begin{array}{c}
\frac{1}{3} \sqrt{3} \\
\frac{1}{3} \sqrt{3} \\
\frac{1}{3} \sqrt{3}
\end{array}\right]\right\} \leftrightarrow 6,\left\{\left[\begin{array}{c}
-\frac{1}{6} \sqrt{6} \\
-\frac{1}{6} \sqrt{6} \\
\frac{1}{3} \sqrt{2} \sqrt{3}
\end{array}\right]\right\} \leftrightarrow 18,\left\{\left[\begin{array}{c}
-\frac{1}{2} \sqrt{2} \\
\frac{1}{2} \sqrt{2} \\
0
\end{array}\right]\right\} \leftrightarrow 24
$$

The matrix $U$ has these as its columns.


Question: City $A$ is located at the origin $(0,0)$ while city $B$ is located at $(300,500)$ where distances are in miles. An airplane flies at 250 miles per hour in still air. This airplane wants to fly from city A to city $B$ but the wind is blowing in the direction of the positive y axis at a speed of 50 miles per hour. Find a unit vector such that if the plane heads in this direction, it will end up at city B having flown the shortest possible distance. How long will it take to get there?

Answer: Therefore, it takes $\frac{583.1}{291.55}=2$ hours.


Question:
Consider the following augmented matrix in which $*$ denotes an arbitrary number and denotes a nonzero number. Determine whether the given augmented matrix is consistent. If consistent, is the solution unique?
$$
\left[\begin{array}{ccccc|c}
\mathbf{\square} & * & * & * & * & * \\
0 & \mathbf{\square} & * & * & 0 & * \\
0 & 0 & \mathbf{\square} & * & * & * \\
0 & 0 & 0 & 0 & \mathbf{\square} & *
\end{array}\right]
$$

Answer: The solution exists but is not unique.


Question: A large dog drags a sled for 300 feet along the ground by pulling on a rope which is 45 degrees from the horizontal with a force of 20 pounds. How much work does this force do?

Answer: $20\left(\cos \frac{\pi}{4}\right) 300=4242.6$


Question:
Find the eigenvalues and an orthonormal basis of eigenvectors for $A$. Diagonalize $A$ by finding an orthogonal matrix $U$ and a diagonal matrix $D$ such that $U^{T} A U=D$.
$$
A=\left[\begin{array}{ccc}
-\frac{5}{3} & \frac{1}{15} \sqrt{6} \sqrt{5} & \frac{8}{15} \sqrt{5} \\
\frac{1}{15} \sqrt{6} \sqrt{5} & -\frac{14}{5} & -\frac{1}{15} \sqrt{6} \\
\frac{8}{15} \sqrt{5} & -\frac{1}{15} \sqrt{6} & \frac{7}{15}
\end{array}\right]
$$
Hint: The eigenvalues are $-3,-2,1$.

Answer:
eigenvectors:

$$
\left\{\left[\begin{array}{c}
\frac{1}{6} \sqrt{6} \\
0 \\
\frac{1}{6} \sqrt{5} \sqrt{6}
\end{array}\right]\right\} \leftrightarrow 1,\left\{\left[\begin{array}{c}
-\frac{1}{3} \sqrt{2} \sqrt{3} \\
-\frac{1}{5} \sqrt{5} \\
\frac{1}{15} \sqrt{2} \sqrt{15}
\end{array}\right]\right\} \leftrightarrow-2,\left\{\left[\begin{array}{c}
-\frac{1}{6} \sqrt{6} \\
\frac{2}{5} \sqrt{5} \\
\frac{1}{30} \sqrt{30}
\end{array}\right]\right\} \leftrightarrow-3
$$

These vectors are the columns of $U$.


Question:
Find the eigenvalues and an orthonormal basis of eigenvectors for $A$. Diagonalize $A$ by finding an orthogonal matrix $U$ and a diagonal matrix $D$ such that $U^{T} A U=D$.
$$
A=\left[\begin{array}{lll}
2 & 0 & 0 \\
0 & 5 & 1 \\
0 & 1 & 5
\end{array}\right]
$$

Answer:
The eigenvectors and eigenvalues are:

$$
\left\{\left[\begin{array}{c}
1 \\
0 \\
0
\end{array}\right]\right\} \leftrightarrow 2,\left\{\left[\begin{array}{c}
0 \\
-\frac{1}{2} \sqrt{2} \\
\frac{1}{2} \sqrt{2}
\end{array}\right]\right\} \leftrightarrow 4,\left\{\left[\begin{array}{c}
0 \\
\frac{1}{2} \sqrt{2} \\
\frac{1}{2} \sqrt{2}
\end{array}\right]\right\} \leftrightarrow 6
$$

These vectors are the columns of $U$.


Question: Find the solution to the system of equations, $65 x+84 y+16 z=546,81 x+105 y+20 z=682$, and $84 x+110 y+21 z=713$.

Answer: Solution is: $[x=2, y=4, z=5]$


Question: Decide whether $\vec{v}=\left[\begin{array}{l}4 \\ 4 \\ 4\end{array}\right]$ is a linear combination of the vectors $\vec{u}_{1}=\left[\begin{array}{r}3 \\ 1 \\ -1\end{array}\right]$ and $\vec{u}_{2}=\left[\begin{array}{r}2 \\ -2 \\ 1\end{array}\right]$.

Answer:
The system
$$
\left[\begin{array}{l}
4 \\
4 \\
4
\end{array}\right]=a_{1}\left[\begin{array}{r}
3 \\
1 \\
-1
\end{array}\right]+a_{2}\left[\begin{array}{r}
2 \\
-2 \\
1
\end{array}\right]
$$
has no solution.


Question:
A person sets off on a random walk with three possible locations. The Markov matrix of probabilities $A=\left[a_{i j}\right]$ is given by
$$
\left[\begin{array}{lll}
0.1 & 0.3 & 0.7 \\
0.1 & 0.3 & 0.2 \\
0.8 & 0.4 & 0.1
\end{array}\right]
$$
If the walker starts in location 2 , what is the probability of ending back in location 2 at time $n=3$ ?

Answer:
$$
X_{3}=\left[\begin{array}{l}
0.38 \\
0.18 \\
0.44
\end{array}\right]
$$

Therefore the probability of ending up back in location 2 is 0.18 .


Question: A girl drags a sled for 200 feet along the ground by pulling on a rope which is 30 degrees from the horizontal with a force of 20 pounds. How much work does this force do?

Answer: $20 \cos \left(\frac{\pi}{6}\right) 200=3464.1$


Question: Find the area of the triangle determined by the three points, $(1,2,3),(2,3,4)$ and $(3,4,5)$. Did something interesting happen here? What does it mean geometrically?

Answer: $\left[\begin{array}{lll}1 & 1 & 1\end{array}\right] \times\left[\begin{array}{lll}2 & 2 & 2\end{array}\right]=\left[\begin{array}{lll}0 & 0 & 0\end{array}\right]$. The area is 0. It means the three points are on the same line.


Question: Suppose $\vec{u}, \vec{v}$, and $\vec{w}$ are three vectors whose components are all integers. Can you conclude the volume of the parallelepiped determined from these three vectors will always be an integer?

Answer: Yes. It will involve the sum of product of integers and so it will be an integer.


Question:
Find the inverse, if it exists, of the matrix
$$
A=\left[\begin{array}{ccc}
e^{t} & \cos t & \sin t \\
e^{t} & -\sin t & \cos t \\
e^{t} & -\cos t & -\sin t
\end{array}\right]
$$

Answer:
$$
\begin{aligned}
& {\left[\begin{array}{ccc}
e^{t} & \cos t & \sin t \\
e^{t} & -\sin t & \cos t \\
e^{t} & -\cos t & -\sin t
\end{array}\right]^{-1} } \\
= & {\left[\begin{array}{ccc}
\frac{1}{2} e^{-t} & 0 & \frac{1}{2} e^{-t} \\
\frac{1}{2} \cos t+\frac{1}{2} \sin t & -\sin t & \frac{1}{2} \sin t-\frac{1}{2} \cos t \\
\frac{1}{2} \sin t-\frac{1}{2} \cos t & \cos t & -\frac{1}{2} \cos t-\frac{1}{2} \sin t
\end{array}\right] }
\end{aligned}
$$


Question: Simplify $\|\vec{u} \times \vec{v}\|^{2}+(\vec{u} \bullet \vec{v})^{2}-\|\vec{u}\|^{2}\|\vec{v}\|^{2}$.

Answer:
$$
\begin{aligned}
\|\vec{u} \times \vec{v}\|^{2} & =\varepsilon_{i j k} u_{j} v_{k} \varepsilon_{i r s} u_{r} v_{s}=\left(\delta_{j r} \delta_{k s}-\delta_{k r} \delta_{j s}\right) u_{r} v_{s} u_{j} v_{k} \\
& =u_{j} v_{k} u_{j} v_{k}-u_{k} v_{j} u_{j} v_{k}=\|\vec{u}\|^{2}\|\vec{v}\|^{2}-(\vec{u} \bullet \vec{v})^{2}
\end{aligned}
$$
It follows that the expression reduces to 0. You can also do the following.
$$
\begin{aligned}
\|\vec{u} \times \vec{v}\|^{2} & =\|\vec{u}\|^{2}\|\vec{v}\|^{2} \sin ^{2} \theta \\
& =\|\vec{u}\|^{2}\|\vec{v}\|^{2}\left(1-\cos ^{2} \theta\right) \\
& =\|\vec{u}\|^{2}\|\vec{v}\|^{2}-\|\vec{u}\|^{2}\|\vec{v}\|^{2} \cos ^{2} \theta \\
& =\|\vec{u}\|^{2}\|\vec{v}\|^{2}-(\vec{u} \bullet \vec{v})^{2}
\end{aligned}
$$
which implies the expression equals 0.


Question: Find $-3\left[\begin{array}{r}5 \\ -1 \\ 2 \\ -3\end{array}\right]+5\left[\begin{array}{r}-8 \\ 2 \\ -3 \\ 6\end{array}\right]$.

Answer: $\left[\begin{array}{r}-55 \\ 13 \\ -21 \\ 39\end{array}\right]$


Question:
Let
$$
A=\left[\begin{array}{lll}
1 & 0 & 3 \\
2 & 3 & 4 \\
1 & 0 & 2
\end{array}\right]
$$
Find $A^{-1}$ if possible. If $A^{-1}$ does not exist, explain why.

Answer: $\left[\begin{array}{lll}1 & 0 & 3 \\ 2 & 3 & 4 \\ 1 & 0 & 2\end{array}\right]^{-1}=\left[\begin{array}{rrr}-2 & 0 & 3 \\ 0 & \frac{1}{3} & -\frac{2}{3} \\ 1 & 0 & -1\end{array}\right]$


Question:
Find the eigenvalues and an orthonormal basis of eigenvectors for $A$. Diagonalize $A$ by finding an orthogonal matrix $U$ and a diagonal matrix $D$ such that $U^{T} A U=D$.
$$
A=\left[\begin{array}{rrr}
13 & 1 & 4 \\
1 & 13 & 4 \\
4 & 4 & 10
\end{array}\right]
$$
Hint: Two eigenvalues are 12 and 18.

Answer:
The eigenvectors and eigenvalues are:

$$
\left\{\left[\begin{array}{c}
-\frac{1}{6} \sqrt{6} \\
-\frac{1}{6} \sqrt{6} \\
\frac{1}{3} \sqrt{2} \sqrt{3}
\end{array}\right]\right\} \leftrightarrow 6,\left\{\left[\begin{array}{c}
-\frac{1}{2} \sqrt{2} \\
\frac{1}{2} \sqrt{2} \\
0
\end{array}\right]\right\} \leftrightarrow 12,\left\{\left[\begin{array}{c}
\frac{1}{3} \sqrt{3} \\
\frac{1}{3} \sqrt{3} \\
\frac{1}{3} \sqrt{3}
\end{array}\right]\right\} \leftrightarrow 18 .
$$

The matrix $U$ has these as its columns.


Question:
An operation is done to get from the first matrix to the second. Identify what was done and tell how it will affect the value of the determinant.

$$
\left[\begin{array}{ll}
a & b \\
c & d
\end{array}\right] \rightarrow \cdots \rightarrow\left[\begin{array}{ll}
c & d \\
a & b
\end{array}\right]
$$

Answer: In this case two rows were switched and so the resulting determinant is -1 times the first.


Question:
Find the following determinant by expanding along the second row and first column.

$$
\left|\begin{array}{lll}
1 & 2 & 1 \\
2 & 1 & 3 \\
2 & 1 & 1
\end{array}\right|
$$

Answer:
$$
\left|\begin{array}{lll}
1 & 2 & 1 \\
2 & 1 & 3 \\
2 & 1 & 1
\end{array}\right|=6
$$


Question:
Consider the following augmented matrix in which $*$ denotes an arbitrary number and denotes a nonzero number. Determine whether the given augmented matrix is consistent. If consistent, is the solution unique?
$$
\left[\begin{array}{lllll|l}
\mathbf{\square} & * & * & * & * & * \\
0 & \mathbf{\square} & * & * & 0 & * \\
0 & 0 & 0 & 0 & \mathbf{\square} & 0 \\
0 & 0 & 0 & 0 & * & \mathbf{\square}
\end{array}\right]
$$

Answer: There might be a solution. If so, there are infinitely many.


Question:
An operation is done to get from the first matrix to the second. Identify what was done and tell how it will affect the value of the determinant.

$$
\left[\begin{array}{ll}
a & b \\
c & d
\end{array}\right] \rightarrow \cdots \rightarrow\left[\begin{array}{cc}
a & b \\
a+c & b+d
\end{array}\right]
$$

Answer: The determinant is unchanged. It was just the first row added to the second.


Question:
Write the solution set of the following system as a linear combination of vectors
$$
\left[\begin{array}{rrrr}
1 & 0 & 1 & 1 \\
1 & -1 & 1 & 0 \\
3 & -1 & 3 & 2 \\
3 & 3 & 0 & 3
\end{array}\right]\left[\begin{array}{l}
x \\
y \\
z \\
w
\end{array}\right]=\left[\begin{array}{l}
0 \\
0 \\
0 \\
0
\end{array}\right]
$$

Answer: Solution is: $\left[\begin{array}{c}0 \\ -\hat{t} \\ -\hat{t} \\ \hat{t}\end{array}\right], \hat{t} \in \mathbb{R}$


Question:
Let $V=\mathbb{R}^{3}$ and let

$$
W=\operatorname{span}\left\{\left[\begin{array}{l}
1 \\
1 \\
1
\end{array}\right],\left[\begin{array}{r}
-1 \\
2 \\
-1
\end{array}\right]\right\}
$$

Extend this basis of $W$ to a basis of $V$.

Answer:
There are many possible such extensions, one is (how do we know?):

$$
\left\{\left[\begin{array}{l}
1 \\
1 \\
1
\end{array}\right],\left[\begin{array}{r}
-1 \\
2 \\
-1
\end{array}\right],\left[\begin{array}{l}
0 \\
0 \\
1
\end{array}\right]\right\}
$$


Question: Find the solution to the system of equations, $3 x-y-2 z=3, y-4 z=0$, and $-2 x+y=-2$.

Answer: Solution is: $[x=2 t+1, y=4 t, z=t]$


Question:
An operation is done to get from the first matrix to the second. Identify what was done and tell how it will affect the value of the determinant.

$$
\left[\begin{array}{ll}
a & b \\
c & d
\end{array}\right] \rightarrow \cdots \rightarrow\left[\begin{array}{cc}
a & b \\
2 c & 2 d
\end{array}\right]
$$

Answer: The second row was multiplied by 2 so the determinant of the result is 2 times the original determinant.


Question:
Write the solution set of the following system as a linear combination of vectors
$$
\left[\begin{array}{rrr}
0 & -1 & 2 \\
1 & 0 & 1 \\
1 & -2 & 5
\end{array}\right]\left[\begin{array}{l}
x \\
y \\
z
\end{array}\right]=\left[\begin{array}{l}
0 \\
0 \\
0
\end{array}\right]
$$

Answer: Solution is: $\left[\begin{array}{c}-\hat{t} \\ 2 \hat{t} \\ \hat{t}\end{array}\right], \hat{t} \in \mathbb{R}$


Question: If you have 6 vectors in $\mathbb{R}^{5}$, is it possible they are linearly independent? Explain.

Answer: No. They can't be.


Question:
Find the determinant using row operations to first simplify.

$$
\left|\begin{array}{rrrr}
1 & 4 & 1 & 2 \\
3 & 2 & -2 & 3 \\
-1 & 0 & 3 & 3 \\
2 & 1 & 2 & -2
\end{array}\right|
$$

Answer:
One can row reduce this using only row operation 3 to

$$
\left[\begin{array}{rrrr}
1 & 4 & 1 & 2 \\
0 & -10 & -5 & -3 \\
0 & 0 & 2 & \frac{19}{5} \\
0 & 0 & 0 & -\frac{211}{20}
\end{array}\right]
$$

Thus the determinant is given by

$$
\left|\begin{array}{rrrr}
1 & 4 & 1 & 2 \\
3 & 2 & -2 & 3 \\
-1 & 0 & 3 & 3 \\
2 & 1 & 2 & -2
\end{array}\right|=211
$$


Question: If $n$ is an integer, is it always true that $(\cos \theta-i \sin \theta)^{n}=\cos (n \theta)-i \sin (n \theta)$ ? Explain.

Answer:
Yes, this is true.

$$
\begin{aligned}
(\cos \theta-i \sin \theta)^{n} & =(\cos (-\theta)+i \sin (-\theta))^{n} \\
& =\cos (-n \theta)+i \sin (-n \theta) \\
& =\cos (n \theta)-i \sin (n \theta)
\end{aligned}
$$


Question: Find the area of the parallelogram determined by the vectors $\left[\begin{array}{l}1 \\ 0 \\ 3\end{array}\right],\left[\begin{array}{r}4 \\ -2 \\ 1\end{array}\right]$.

Answer: $\left[\begin{array}{l}1 \\ 0 \\ 3\end{array}\right] \times\left[\begin{array}{r}4 \\ -2 \\ 1\end{array}\right]=\left[\begin{array}{r}6 \\ 11 \\ -2\end{array}\right]$. The area is $\sqrt{36+121+4}=\sqrt{161}$


Question: Find the matrix for $T(\vec{w})=\operatorname{proj}_{\vec{v}}(\vec{w})$ where $\vec{v}=\left[\begin{array}{lll}1 & 5 & 3\end{array}\right]^{T}$.

Answer:
$$
\frac{1}{35}\left[\begin{array}{rrr}
1 & 5 & 3 \\
5 & 25 & 15 \\
3 & 15 & 9
\end{array}\right]
$$


Question: Suppose $A \vec{x}=\vec{b}$ has a solution. Explain why the solution is unique precisely when $A \vec{x}=\overrightarrow{0}$ has only the trivial solution.

Answer: If not, then there would be a infintely many solutions to $A \vec{x}=\overrightarrow{0}$ and each of these added to a solution to $A \vec{x}=\vec{b}$ would be a solution to $A \vec{x}=\vec{b}$.


Question:
Find the eigenvalues and an orthonormal basis of eigenvectors for $A$.
$$
A=\left[\begin{array}{rrr}
4 & 1 & -2 \\
1 & 4 & -2 \\
-2 & -2 & 7
\end{array}\right]
$$
Hint: One eigenvalue is 3.

Answer:
The eigenvectors and eigenvalues are:

$$
\left\{\frac{1}{\sqrt{2}}\left[\begin{array}{r}
-1 \\
1 \\
0
\end{array}\right], \frac{1}{\sqrt{3}}\left[\begin{array}{l}
1 \\
1 \\
1
\end{array}\right]\right\} \leftrightarrow 3,\left\{\frac{1}{\sqrt{6}}\left[\begin{array}{r}
-1 \\
-1 \\
2
\end{array}\right]\right\} \leftrightarrow 9
$$


Question:
Write the solution set of the following system as a linear combination of vectors
$$
\left[\begin{array}{lll}
1 & -1 & 2 \\
1 & -2 & 1 \\
3 & -4 & 5
\end{array}\right]\left[\begin{array}{l}
x \\
y \\
z
\end{array}\right]=\left[\begin{array}{l}
0 \\
0 \\
0
\end{array}\right]
$$

Answer: Solution is: $\left[\begin{array}{r}-3 \hat{t} \\ -\hat{t} \\ \hat{t}\end{array}\right], \hat{t}_{3} \in \mathbb{R}$. A basis for the solution space is $\left[\begin{array}{r}-3 \\ -1 \\ 1\end{array}\right]$


Question: If $A$ is an invertible $n \times n$ matrix, compare the eigenvalues of $A$ and $A^{-1}$. More generally, for $m$ an arbitrary integer, compare the eigenvalues of $A$ and $A^{m}$.

Answer: $A^{m} X=\lambda^{m} X$ for any integer. In the case of $-1, A^{-1} \lambda X=A A^{-1} X=X$ so $A^{-1} X=\lambda^{-1} X$. Thus the eigenvalues of $A^{-1}$ are just $\lambda^{-1}$ where $\lambda$ is an eigenvalue of $A$.


Question: What does it mean geometrically if the box product of three vectors gives zero?

Answer: It means that if you place them so that they all have their tails at the same point, the three will lie in the same plane.


Question:
Find the volume of the parallelepiped determined by the vectors $\left[\begin{array}{r}1 \\ -7 \\ -5\end{array}\right]$,
$\left[\begin{array}{r}1 \\ -2 \\ -6\end{array}\right]$, and $\left[\begin{array}{l}3 \\ 2 \\ 3\end{array}\right]$

Answer: $\left|\begin{array}{rrr}1 & -7 & -5\end{array}\right|=113$


Question:
Find the eigenvalues and an orthonormal basis of eigenvectors for $A$. Diagonalize $A$ by finding an orthogonal matrix $U$ and a diagonal matrix $D$ such that $U^{T} A U=D$.
$$
A=\left[\begin{array}{ccc}
\frac{4}{3} & \frac{1}{3} \sqrt{3} \sqrt{2} & \frac{1}{3} \sqrt{2} \\
\frac{1}{3} \sqrt{3} \sqrt{2} & 1 & -\frac{1}{3} \sqrt{3} \\
\frac{1}{3} \sqrt{2} & -\frac{1}{3} \sqrt{3} & \frac{5}{3}
\end{array}\right]
$$
Hint: The eigenvalues are 0,2,2 where 2 is listed twice because it is a root of multiplicity 2.

Answer:
The eigenvectors and eigenvalues are:

$$
\left\{\left[\begin{array}{c}
-\frac{1}{5} \sqrt{2} \sqrt{5} \\
\frac{1}{5} \sqrt{3} \sqrt{5} \\
\frac{1}{5} \sqrt{5}
\end{array}\right]\right\} \leftrightarrow 0,\left\{\left[\begin{array}{c}
\frac{1}{3} \sqrt{3} \\
0 \\
\frac{1}{3} \sqrt{2} \sqrt{3}
\end{array}\right],\left[\begin{array}{c}
\frac{1}{5} \sqrt{2} \sqrt{5} \\
\frac{1}{5} \sqrt{3} \sqrt{5} \\
-\frac{1}{5} \sqrt{5}
\end{array}\right]\right\} \leftrightarrow 2
$$

The columns are these vectors.


Question: Find the matrix for $T(\vec{w})=\operatorname{proj}_{\vec{v}}(\vec{w})$ where $\vec{v}=\left[\begin{array}{lll}1 & 0 & 3\end{array}\right]^{T}$.

Answer:
$$
\frac{1}{10}\left[\begin{array}{lll}
1 & 0 & 3 \\
0 & 0 & 0 \\
3 & 0 & 9
\end{array}\right]
$$


Question: Consider functions defined on $\{1,2, \cdots, n\}$ having values in $\mathbb{R}$. Explain how, if $V$ is the set of all such functions, $V$ can be considered as $\mathbb{R}^{n}$.

Answer: Let $f(i)$ be the $i^{t h}$ component of a vector $\vec{x} \in \mathbb{R}^{n}$. Thus a typical element in $\mathbb{R}^{n}$ is $(f(1), \cdots, f(n))$.


Question: The steady state temperature, $u$, of a plate solves Laplace's equation, $\Delta u=0$. One way to approximate the solution is to divide the plate into a square mesh and require the temperature at each node to equal the average of the temperature at the four adjacent nodes. In the following picture, the numbers represent the observed temperature at the indicated nodes. Find the temperature at the interior nodes, indicated by $x, y, z$, and $w$. One of the equations is $z=\frac{1}{4}(10+0+w+x)$.

Answer: Solution is: $[w=15, x=15, y=20, z=10]$


Question: Find $\operatorname{proj}_{\vec{v}}(\vec{w})$ where $\vec{w}=\left[\begin{array}{r}1 \\ 2 \\ -2 \\ 1\end{array}\right]$ and $\vec{v}=\left[\begin{array}{l}1 \\ 2 \\ 3 \\ 0\end{array}\right]$.

Answer: $\overrightarrow{\vec{u} \bullet \vec{v}} \overrightarrow{\vec{u}} \vec{u}=\frac{\left[\begin{array}{llll}1 & 2 & -2 & 1\end{array}\right]^{T} \bullet\left[\begin{array}{llll}1 & 2 & 3 & 0\end{array}\right]^{T}}{1+4+9}\left[\begin{array}{l}1 \\ 2 \\ 3 \\ 0\end{array}\right]=\left[\begin{array}{r}-\frac{1}{14} \\ -\frac{1}{7} \\ -\frac{3}{14} \\ 0\end{array}\right]$


Question:
Find the determinant using row operations to first simplify.

$$
\left|\begin{array}{rrrr}
1 & 2 & 1 & 2 \\
3 & 1 & -2 & 3 \\
-1 & 0 & 3 & 1 \\
2 & 3 & 2 & -2
\end{array}\right|
$$

Answer:
One can row reduce this using only row operation 3 to

$$
\left[\begin{array}{rrrr}
1 & 2 & 1 & 2 \\
0 & -5 & -5 & -3 \\
0 & 0 & 2 & \frac{9}{5} \\
0 & 0 & 0 & -\frac{63}{10}
\end{array}\right]
$$

and therefore, the determinant is -63.

$$
\left|\begin{array}{rrrr}
1 & 2 & 1 & 2 \\
3 & 1 & -2 & 3 \\
-1 & 0 & 3 & 1 \\
2 & 3 & 2 & -2
\end{array}\right|=63
$$


Question: Is $\vec{u} \times(\vec{v} \times \vec{w})=(\vec{u} \times \vec{v}) \times \vec{w}$ ? What is the meaning of $\vec{u} \times \vec{v} \times \vec{w}$ ? Explain. Hint: Try $(\vec{i} \times \vec{j}) \times \vec{k}$

Answer: $(\vec{i} \times \vec{j}) \times \vec{j}=\vec{k} \times \vec{j}=-\vec{i}$. However, $\vec{i} \times(\vec{j} \times \vec{j})=\overrightarrow{0}$ and so the cross product is not associative.


Question:
Find the following determinant by expanding along the first column and third row.

$$
\left|\begin{array}{lll}
1 & 2 & 1 \\
1 & 0 & 1 \\
2 & 1 & 1
\end{array}\right|
$$

Answer:
$$
\left|\begin{array}{lll}
1 & 2 & 1 \\
1 & 0 & 1 \\
2 & 1 & 1
\end{array}\right|=2
$$


Question: Find the matrix for $T(\vec{w})=\operatorname{proj}_{\vec{v}}(\vec{w})$ where $\vec{v}=\left[\begin{array}{lll}1 & 5 & 3\end{array}\right]^{T}$.

Answer:
$$
\frac{1}{35}\left[\begin{array}{rrr}
1 & 5 & 3 \\
5 & 25 & 15 \\
3 & 15 & 9
\end{array}\right]
$$


Question:
Find the solution of the system whose augmented matrix is
$$
\left[\begin{array}{lll|l}
1 & 1 & 0 & 1 \\
1 & 0 & 4 & 2
\end{array}\right]
$$

Answer: The solution is $z=t, y=4 t, x=2-4 t$.


Question:
Write the solution set of the following system as a linear combination of vectors
$$
\left[\begin{array}{lll}
0 & -1 & 2 \\
1 & -2 & 1 \\
1 & -4 & 5
\end{array}\right]\left[\begin{array}{l}
x \\
y \\
z
\end{array}\right]=\left[\begin{array}{l}
0 \\
0 \\
0
\end{array}\right]
$$

Answer: Solution is: $\left[\begin{array}{c}3 \hat{t} \\ 2 \hat{t} \\ \hat{t}\end{array}\right]$, A basis is $\left[\begin{array}{l}3 \\ 2 \\ 1\end{array}\right]$


Question: Find the solution to the system of equations, $8 x+2 y+3 z=-3,8 x+3 y+3 z=-1$, and $4 x+y+3 z=-9$.

Answer: Solution is: $[x=1, y=2, z=-5]$


Question: Find $\left[\begin{array}{l}1 \\ 2 \\ 3 \\ 4\end{array}\right] \bullet\left[\begin{array}{l}2 \\ 0 \\ 1 \\ 3\end{array}\right]$.

Answer: $\left[\begin{array}{l}1 \\ 2 \\ 3 \\ 4\end{array}\right] \bullet\left[\begin{array}{l}2 \\ 0 \\ 1 \\ 3\end{array}\right]=17$


Question: A boy drags a sled for 100 feet along the ground by pulling on a rope which is 20 degrees from the horizontal with a force of 40 pounds. How much work does this force do?

Answer: $40 \cos \left(\frac{20}{180} \pi\right) 100=3758.8$


Question:
Find $h$ such that
$$
\left[\begin{array}{ll|l}
2 & h & 4 \\
3 & 6 & 7
\end{array}\right]
$$
is the augmented matrix of an inconsistent system.

Answer: $h=4$


Question:
Let
$$
A=\left[\begin{array}{lll}
1 & 2 & 3 \\
2 & 1 & 4 \\
1 & 0 & 2
\end{array}\right]
$$
Find $A^{-1}$ if possible. If $A^{-1}$ does not exist, explain why.

Answer: $\left[\begin{array}{lll}1 & 2 & 3 \\ 2 & 1 & 4 \\ 1 & 0 & 2\end{array}\right]^{-1}=\left[\begin{array}{rrr}-2 & 4 & -5 \\ 0 & 1 & -2 \\ 1 & -2 & 3\end{array}\right]$


Question:
Find the eigenvalues and an orthonormal basis of eigenvectors for $A$. Diagonalize $A$ by finding an orthogonal matrix $U$ and a diagonal matrix $D$ such that $U^{T} A U=D$.
$$
A=\left[\begin{array}{ccc}
3 & 0 & 0 \\
0 & \frac{3}{2} & \frac{1}{2} \\
0 & \frac{1}{2} & \frac{3}{2}
\end{array}\right]
$$

Answer:
The eigenvectors and eigenvalues are: $\left\{\left[\begin{array}{c}0 \\ -\frac{1}{2} \sqrt{2} \\ \frac{1}{2} \sqrt{2}\end{array}\right]\right\} \leftrightarrow 1,\left\{\left[\begin{array}{c}0 \\ \frac{1}{2} \sqrt{2} \\ \frac{1}{2} \sqrt{2}\end{array}\right]\right\} \leftrightarrow 2,\left\{\left[\begin{array}{l}1 \\ 0 \\ 0\end{array}\right]\right\} \leftrightarrow 3$.

These vectors are the columns of the matrix $U$.


Question:
Find the determinant using row operations to first simplify.

$$
\left|\begin{array}{rrr}
1 & 2 & 1 \\
2 & 3 & 2 \\
-4 & 1 & 2
\end{array}\right|
$$

Answer:
$$
\left|\begin{array}{rrr}
1 & 2 & 1 \\
2 & 3 & 2 \\
-4 & 1 & 2
\end{array}\right|=-6
$$


Question:
Find the determinant using row operations to first simplify.

$$
\left|\begin{array}{rrr}
2 & 1 & 3 \\
2 & 4 & 2 \\
1 & 4 & -5
\end{array}\right|
$$

Answer:
$$
\left|\begin{array}{rrr}
2 & 1 & 3 \\
2 & 4 & 2 \\
1 & 4 & -5
\end{array}\right|=-32
$$


Question: Find the solution to the system of equations, $3 x-y+4 z=6, y+8 z=0$, and $-2 x+y=-4$.

Answer: Solution is: $[x=2-4 t, y=-8 t, z=t]$


Question:
Consider the matrix
$$
A=\left[\begin{array}{ccc}
1 & 0 & 0 \\
0 & \cos t & -\sin t \\
0 & \sin t & \cos t
\end{array}\right]
$$
Does there exist a value of t for which this matrix fails to have an inverse? Explain.

Answer: No. It has a nonzero determinant for all $t$


Question:
Find the eigenvalues and an orthonormal basis of eigenvectors for the matrix
$$
A=\left[\begin{array}{ccc}
\frac{1}{3} & \frac{1}{6} \sqrt{3} \sqrt{2} & \frac{1}{6} \sqrt{3} \sqrt{6} \\
\frac{1}{6} \sqrt{3} \sqrt{2} & \frac{3}{2} & \frac{1}{12} \sqrt{2} \sqrt{6} \\
\frac{1}{6} \sqrt{3} \sqrt{6} & \frac{1}{12} \sqrt{2} \sqrt{6} & \frac{7}{6}
\end{array}\right]
$$
Hint: The eigenvalues are 1,2,-2.

Answer:
eigenvectors:

$$
\left\{\left[\begin{array}{c}
-\frac{1}{3} \sqrt{3} \\
\frac{1}{2} \sqrt{2} \\
\frac{1}{6} \sqrt{6}
\end{array}\right]\right\} \leftrightarrow 1,\left\{\left[\begin{array}{c}
\frac{1}{3} \sqrt{3} \\
0 \\
\frac{1}{3} \sqrt{2} \sqrt{3}
\end{array}\right]\right\} \leftrightarrow-2,\left\{\left[\begin{array}{c}
\frac{1}{3} \sqrt{3} \\
\frac{1}{2} \sqrt{2} \\
-\frac{1}{6} \sqrt{6}
\end{array}\right]\right\} \leftrightarrow 2 .
$$

Then the columns of $U$ are these vectors


Question: Find the solution to the system of equations, $9 x-2 y+4 z=-17,13 x-3 y+6 z=-25$, and $-2 x-z=3$.

Answer: Solution is: $[x=-1, y=2, z=-1]$


Question: Give the complete solution to $x^{4}+16=0$.

Answer:
Solution is:

$$
(1-i) \sqrt{2},-(1+i) \sqrt{2},-(1-i) \sqrt{2},(1+i) \sqrt{2}
$$


Question: If $A$ is an $n \times n$ matrix and $c$ is a nonzero constant, compare the eigenvalues of $A$ and $c A$.

Answer: Say $A X=\lambda X$. Then $c A X=c \lambda X$ and so the eigenvalues of $c A$ are just $c \lambda$ where $\lambda$ is an eigenvalue of $A$.


Question:
Find the eigenvalues and an orthonormal basis of eigenvectors for $A$. Diagonalize $A$ by finding an orthogonal matrix $U$ and a diagonal matrix $D$ such that $U^{T} A U=D$.
$$
A=\left[\begin{array}{ccc}
1 & \frac{1}{6} \sqrt{3} \sqrt{2} & \frac{1}{6} \sqrt{3} \sqrt{6} \\
\frac{1}{6} \sqrt{3} \sqrt{2} & \frac{3}{2} & \frac{1}{12} \sqrt{2} \sqrt{6} \\
\frac{1}{6} \sqrt{3} \sqrt{6} & \frac{1}{12} \sqrt{2} \sqrt{6} & \frac{1}{2}
\end{array}\right]
$$
Hint: The eigenvalues are 2, 1,0.

Answer:
The eigenvectors and eigenvalues are:

$$
\left\{\left[\begin{array}{c}
-\frac{1}{3} \sqrt{3} \\
0 \\
\frac{1}{3} \sqrt{2} \sqrt{3}
\end{array}\right]\right\} \leftrightarrow 0,\left\{\left[\begin{array}{c}
\frac{1}{3} \sqrt{3} \\
-\frac{1}{2} \sqrt{2} \\
\frac{1}{6} \sqrt{6}
\end{array}\right]\right\} \leftrightarrow 1,\left\{\left[\begin{array}{c}
\frac{1}{3} \sqrt{3} \\
\frac{1}{2} \sqrt{2} \\
\frac{1}{6} \sqrt{6}
\end{array}\right]\right\} \leftrightarrow 2
$$

The columns are these vectors.


Question: Find the solution to the system of equations, $-19 x+8 y=-108,-71 x+30 y=-404$, $-2 x+y=-12,4 x+z=14$.

Answer: Solution is: $[x=4, y=-4, z=-2]$


Question:
Find the eigenvalues and eigenvectors of the matrix
$$
\left[\begin{array}{rrr}
-13 & -28 & 28 \\
4 & 9 & -8 \\
-4 & -8 & 9
\end{array}\right]
$$
One eigenvalue is 3. Diagonalize if possible.

Answer:
The eigenvectors and eigenvalues are:

$$
\left\{\left[\begin{array}{l}
2 \\
0 \\
1
\end{array}\right]\right\} \leftrightarrow 1,\left\{\left[\begin{array}{c}
-2 \\
1 \\
0
\end{array}\right]\right\} \leftrightarrow 1,\left\{\left[\begin{array}{c}
7 \\
-2 \\
2
\end{array}\right]\right\} \leftrightarrow 3
$$

The matrix $P$ needed to diagonalize the above matrix is

$$
\left[\begin{array}{rrr}
2 & -2 & 7 \\
0 & 1 & -2 \\
1 & 0 & 2
\end{array}\right]
$$

and the diagonal matrix $D$ is

$$
\left[\begin{array}{lll}
1 & 0 & 0 \\
0 & 1 & 0 \\
0 & 0 & 3
\end{array}\right]
$$


Question:
An operation is done to get from the first matrix to the second. Identify what was done and tell how it will affect the value of the determinant.

$$
\left[\begin{array}{ll}
a & b \\
c & d
\end{array}\right] \rightarrow \cdots \rightarrow\left[\begin{array}{ll}
b & a \\
d & c
\end{array}\right]
$$

Answer: In this case the two columns were switched so the determinant of the second is -1 times the determinant of the first.


Question: The wind blows from West to East at a speed of 50 miles per hour and an airplane which travels at 400 miles per hour in still air is heading North West. What is the velocity of the airplane relative to the ground? What is the component of this velocity in the direction North?

Answer: The velocity is the sum of two vectors. $50 \vec{i}+\frac{300}{\sqrt{2}}(\vec{i}+\vec{j})=\left(50+\frac{300}{\sqrt{2}}\right) \vec{i}+\frac{300}{\sqrt{2}} \vec{j}$. The component in the direction of North is then $\frac{300}{\sqrt{2}}=150 \sqrt{2}$ and the velocity relative to the ground is $\left(50+\frac{300}{\sqrt{2}}\right) \vec{i}+\frac{300}{\sqrt{2}} \vec{j}$


Question:
Find the eigenvalues and an orthonormal basis of eigenvectors for $A$.
$$
A=\left[\begin{array}{rrr}
11 & -1 & -4 \\
-1 & 11 & -4 \\
-4 & -4 & 14
\end{array}\right]
$$
Hint: Two eigenvalues are 12 and 18.

Answer:
The eigenvectors and eigenvalues are:

$$
\left\{\frac{1}{\sqrt{3}}\left[\begin{array}{l}
1 \\
1 \\
1
\end{array}\right]\right\} \leftrightarrow 6,\left\{\frac{1}{\sqrt{2}}\left[\begin{array}{r}
-1 \\
1 \\
0
\end{array}\right]\right\} \leftrightarrow 12,\left\{\frac{1}{\sqrt{6}}\left[\begin{array}{r}
-1 \\
-1 \\
2
\end{array}\right]\right\} \leftrightarrow 18
$$


Question: Suppose a system of equations has fewer equations than variables and you have found a solution to this system of equations. Is it possible that your solution is the only one? Explain.

Answer: No. Consider $x+y+z=2$ and $x+y+z=1$.


Question: If you have 5 vectors in $\mathbb{R}^{5}$ and the vectors are linearly independent, can it always be concluded they span $\mathbb{R}^{5}$ ?

Answer: Yes. If not, there would exist a vector not in the span. But then you could add in this vector and obtain a linearly independent set of vectors with more vectors than a basis.


Question:
Write the solution set of the following system as a linear combination of vectors
$$
\left[\begin{array}{llll}
1 & 1 & 0 & 1 \\
2 & 1 & 1 & 2 \\
1 & 0 & 1 & 1 \\
0 & 0 & 0 & 0
\end{array}\right]\left[\begin{array}{l}
x \\
y \\
z \\
w
\end{array}\right]=\left[\begin{array}{l}
0 \\
0 \\
0 \\
0
\end{array}\right]
$$

Answer:
Solution is: $\left[\begin{array}{c}-s-t \\ s \\ s \\ t\end{array}\right], s, t \in \mathbb{R}$. A basis is

$$
\left\{\left[\begin{array}{r}
-1 \\
1 \\
1 \\
0
\end{array}\right],\left[\begin{array}{r}
-1 \\
0 \\
0 \\
1
\end{array}\right]\right\}
$$


Question: Find the point $\left(x_{1}, y_{1}\right)$ which lies on both lines, $x+3 y=1$ and $4 x-y=3$.

Answer: Solution is: $\left[x=\frac{10}{13}, y=\frac{1}{13}\right]$


Question: Find the four fourth roots of 16.

Answer:
Solution is:

$$
(1-i) \sqrt{2},-(1+i) \sqrt{2},-(1-i) \sqrt{2},(1+i) \sqrt{2}
$$


Question: Let $M=\left\{\vec{u}=\left(u_{1}, u_{2}, u_{3}, u_{4}\right) \in \mathbb{R}^{4}: \sin \left(u_{1}\right)=1\right\}$. Is $M$ a subspace of $\mathbb{R}^{4}$ ?

Answer: This is not a subspace.


Question: Find the area of the triangle determined by the three points, $(1,0,3),(4,1,0)$ and $(-3,1,1)$.

Answer:
$\left[\begin{array}{r}3 \\ 1 \\ -3\end{array}\right] \times\left[\begin{array}{r}-4 \\ 1 \\ -2\end{array}\right]=\left[\begin{array}{c}1 \\ 18 \\ 7\end{array}\right]$. The area is given by
$$
\frac{1}{2} \sqrt{1+(18)^{2}+49}=\frac{1}{2} \sqrt{374}
$$


Question:
Let $V=\mathbb{R}^{3}$ and let
$$
W=\operatorname{span}(S), \text { where } S=\left\{\left[\begin{array}{r}
1 \\
-1 \\
1
\end{array}\right],\left[\begin{array}{r}
-2 \\
2 \\
-2
\end{array}\right],\left[\begin{array}{r}
-1 \\
1 \\
1
\end{array}\right],\left[\begin{array}{r}
1 \\
-1 \\
3
\end{array}\right]\right\}
$$
Find a basis of $W$ consisting of vectors in $S$.

Answer: In this case $\operatorname{dim}(W)=1$ and a basis for $W$ consisting of vectors in $S$ can be obtained by taking any (nonzero) vector from $S$.


Question: Let $B=\left\{\left[\begin{array}{r}1 \\ -1 \\ 2\end{array}\right],\left[\begin{array}{l}2 \\ 1 \\ 2\end{array}\right],\left[\begin{array}{r}-1 \\ 0 \\ 2\end{array}\right]\right\}$ be a basis of $\mathbb{R}^{3}$ and let $\vec{x}=\left[\begin{array}{r}5 \\ -1 \\ 4\end{array}\right]$ be a vector in $\mathbb{R}^{2}$. Find $C_{B}(\vec{x})$.

Answer: $C_{B}(\vec{x})=\left[\begin{array}{r}2 \\ 1 \\ -1\end{array}\right]$.


Question:
Find the least squares solution to the following system.
$$
\begin{gathered}
x+2 y=1 \\
2 x+3 y=2 \\
3 x+5 y=4
\end{gathered}
$$

Answer: Solution is: $\left[\begin{array}{c}\frac{2}{3} \\ \frac{1}{3}\end{array}\right]$


Question:
Find the angle between the vectors
$$
\vec{u}=\left[\begin{array}{r}
3 \\
-1 \\
-1
\end{array}\right], \vec{v}=\left[\begin{array}{l}
1 \\
4 \\
2
\end{array}\right]
$$

Answer:
$\frac{\left[\begin{array}{lll}3 & -1 & -1\end{array}\right]^{T} \cdot\left[\begin{array}{lll}1 & 4 & 2\end{array}\right]^{T}}{\sqrt{9+1+1} \sqrt{1+16+4}}=\frac{-3}{\sqrt{11} \sqrt{21}}=-0.19739=\cos \theta$
Therefore we need to solve
$$
-0.19739=\cos \theta
$$
Thus $\theta=1.7695$ radians.


Question:
Let $V=\mathbb{R}^{3}$ and let

$$
W=\operatorname{span}(S), \text { where } S=\left\{\left[\begin{array}{r}
1 \\
-1 \\
1
\end{array}\right],\left[\begin{array}{r}
-2 \\
2 \\
-2
\end{array}\right],\left[\begin{array}{r}
-1 \\
1 \\
1
\end{array}\right],\left[\begin{array}{r}
1 \\
-1 \\
3
\end{array}\right]\right\}
$$

Find a basis of $W$ consisting of vectors in $S$.

Answer: In this case $\operatorname{dim}(W)=1$ and a basis for $W$ consisting of vectors in $S$ can be obtained by taking any (nonzero) vector from $S$.


Question: Simplify $(\vec{u} \times \vec{v}) \bullet(\vec{v} \times \vec{w}) \times(\vec{w} \times \vec{z})$.

Answer:
Here $[\vec{v}, \vec{w}, \vec{z}]$ denotes the box product. Consider the cross product term. From the above,
$$
\begin{aligned}
(\vec{v} \times \vec{w}) \times(\vec{w} \times \vec{z}) & =[\vec{v}, \vec{w}, \vec{z}] \vec{w}-[\vec{w}, \vec{w}, \vec{z}] \vec{v} \\
& =[\vec{v}, \vec{w}, \vec{z}] \vec{w}
\end{aligned}
$$
Thus it reduces to
$$
(\vec{u} \times \vec{v}) \bullet[\vec{v}, \vec{w}, \vec{z}] \vec{w}=[\vec{v}, \vec{w}, \vec{z}][\vec{u}, \vec{v}, \vec{w}]
$$


Question: Let $B=\left\{\left[\begin{array}{r}1 \\ -1 \\ 2\end{array}\right],\left[\begin{array}{l}2 \\ 1 \\ 2\end{array}\right],\left[\begin{array}{r}-1 \\ 0 \\ 2\end{array}\right]\right\}$ be a basis of $\mathbb{R}^{3}$ and let $\vec{x}=\left[\begin{array}{r}5 \\ -1 \\ 4\end{array}\right]$ be a vector in $\mathbb{R}^{2}$. Find $C_{B}(\vec{x})$.

Answer: $C_{B}(\vec{x})=\left[\begin{array}{r}2 \\ 1 \\ -1\end{array}\right]$


Question: Consider the system $-5 x+2 y-z=0$ and $-5 x-2 y-z=0$. Both equations equal zero and so $-5 x+2 y-z=-5 x-2 y-z$ which is equivalent to $y=0$. Does it follow that $x$ and $z$ can equal anything? Notice that when $x=1, z=-4$, and $y=0$ are plugged in to the equations, the equations do not equal 0 . Why?

Answer: These are not legitimate row operations. They do not preserve the solution set of the system.


Question: Find $\operatorname{proj}_{\vec{v}}(\vec{w})$ where $\vec{w}=\left[\begin{array}{r}1 \\ 0 \\ -2\end{array}\right]$ and $\vec{v}=\left[\begin{array}{l}1 \\ 2 \\ 3\end{array}\right]$.

Answer: $\frac{\vec{u} \bullet \vec{v}}{\vec{u} \bullet \vec{u}} \vec{u}=\frac{-5}{14}\left[\begin{array}{l}1 \\ 2 \\ 3\end{array}\right]=\left[\begin{array}{r}-\frac{5}{14} \\ -\frac{5}{7} \\ -\frac{15}{14}\end{array}\right]$


Question:
Find the eigenvalues and an orthonormal basis of eigenvectors for the matrix
$$
A=\left[\begin{array}{ccc}
-\frac{1}{2} & -\frac{1}{5} \sqrt{6} \sqrt{5} & \frac{1}{10} \sqrt{5} \\
-\frac{1}{5} \sqrt{6} \sqrt{5} & \frac{7}{5} & -\frac{1}{5} \sqrt{6} \\
\frac{1}{10} \sqrt{5} & -\frac{1}{5} \sqrt{6} & -\frac{9}{10}
\end{array}\right]
$$
Hint: The eigenvalues are $-1,2,-1$ where -1 is listed twice because it has multiplicity 2 as a zero of the characteristic equation.

Answer:
The eigenvectors and eigenvalues are:

$$
\left\{\left[\begin{array}{c}
-\frac{1}{6} \sqrt{6} \\
0 \\
\frac{1}{6} \sqrt{5} \sqrt{6}
\end{array}\right],\left[\begin{array}{c}
\frac{1}{3} \sqrt{2} \sqrt{3} \\
\frac{1}{5} \sqrt{5} \\
\frac{1}{15} \sqrt{2} \sqrt{15}
\end{array}\right]\right\} \leftrightarrow-1,\left\{\left[\begin{array}{c}
\frac{1}{6} \sqrt{6} \\
-\frac{2}{5} \sqrt{5} \\
\frac{1}{30} \sqrt{30}
\end{array}\right]\right\} \leftrightarrow 2 .
$$

The columns of $U$ are these vectors.


Question: Find the area of the parallelogram determined by the vectors $\left[\begin{array}{l}1 \\ 2 \\ 3\end{array}\right],\left[\begin{array}{r}3 \\ -2 \\ 1\end{array}\right]$.

Answer: $\left[\begin{array}{l}1 \\ 2 \\ 3\end{array}\right] \times\left[\begin{array}{r}3 \\ -2 \\ 1\end{array}\right]=\left[\begin{array}{r}8 \\ 8 \\ -8\end{array}\right]$. The area is $8 \sqrt{3}$


Question: An object moves 20 meters in the direction of $\vec{k}+\vec{j}$. There are two forces acting on this object, $\vec{F}_{1}=\vec{i}+\vec{j}+2 \vec{k}$, and $\vec{F}_{2}=\vec{i}+2 \vec{j}-6 \vec{k}$. Find the total work done on the object by the two forces. Hint: You can take the work done by the resultant of the two forces or you can add the work done by each force. Why?

Answer: $\left[\begin{array}{r}2 \\ 3 \\ -4\end{array}\right] \bullet\left[\begin{array}{c}0 \\ \frac{1}{\sqrt{2}} \\ \frac{1}{\sqrt{2}}\end{array}\right] 20=-10 \sqrt{2}$


Question: Suppose $A B=A C$ and $A$ is an invertible $n \times n$ matrix. Does it follow that $B=C$ ? Explain why or why not.

Answer: Yes $B=C$. Multiply $A B=A C$ on the left by $A^{-1}$.


Question: Let $A=\left[\begin{array}{ll}1 & 2 \\ 3 & 4\end{array}\right], B=\left[\begin{array}{ll}1 & 2 \\ 3 & k\end{array}\right]$. Is it possible to choose $k$ such that $A B=B A$ ? If so, what should $k$ equal?

Answer: Solution is: $[k=4]$


Question: Find the solution to the system of equations, $7 x+14 y+15 z=22,2 x+4 y+3 z=5$, and $3 x+6 y+10 z=13$.

Answer: Solution is: $[x=1-2 t, z=1, y=t]$


Question:
Consider the following augmented matrix in which $*$ denotes an arbitrary number and denotes a nonzero number. Determine whether the given augmented matrix is consistent. If consistent, is the solution unique?
$$
\left[\begin{array}{ccc|c}
\mathbf{0} & * & * & * \\
0 & \mathbf{\square} & * & * \\
0 & 0 & \mathbf{\square} & *
\end{array}\right]
$$

Answer: A solution exists and is unique.


Question:
Determine if the system is consistent. If so, is the solution unique?
$$
\begin{gathered}
x+2 y+z-w=2 \\
x-y+z+w=1 \\
2 x+y-z=1 \\
4 x+2 y+z=5
\end{gathered}
$$

Answer: There is no solution.


Question:
Find the eigenvalues and eigenvectors of the matrix
$$
\left[\begin{array}{rrr}
5 & -18 & -32 \\
0 & 5 & 4 \\
2 & -5 & -11
\end{array}\right]
$$
One eigenvalue is 1. Diagonalize if possible.

Answer:
The eigenvalues are $-1,-1,1$. The eigenvectors corresponding to the eigenvalues are:

$$
\left\{\left[\begin{array}{c}
10 \\
-2 \\
3
\end{array}\right]\right\} \leftrightarrow-1,\left\{\left[\begin{array}{c}
7 \\
-2 \\
2
\end{array}\right]\right\} \leftrightarrow 1
$$

Therefore this matrix is not diagonalizable.


Question: Suppose the coefficient matrix of a system of $n$ equations with $n$ variables has the property that every column is a pivot column. Does it follow that the system of equations must have a solution? If so, must the solution be unique? Explain.

Answer: Yes. It has a unique solution.


Question: Let $X=\left[\begin{array}{lll}-1 & -1 & 1\end{array}\right]$ and $Y=\left[\begin{array}{lll}0 & 1 & 2\end{array}\right]$. Find $X^{T} Y$ and $X Y^{T}$ if possible.

Answer: $X^{T} Y=\left[\begin{array}{rrr}0 & -1 & -2 \\ 0 & -1 & -2 \\ 0 & 1 & 2\end{array}\right], X Y^{T}=1$


Question:
Let
$$
A=\left[\begin{array}{lll}
1 & 2 & 1 \\
2 & 1 & 4 \\
4 & 5 & 10
\end{array}\right]
$$
Find $A^{-1}$ if possible. If $A^{-1}$ does not exist, explain why.

Answer: $\left[\begin{array}{rrrr}1 & 2 & 0 & 2 \\ 1 & 1 & 2 & 0 \\ 2 & 1 & -3 & 2 \\ 1 & 2 & 1 & 2\end{array}\right]^{-1}=\left[\begin{array}{rrrr}-1 & \frac{1}{2} & \frac{1}{2} & \frac{1}{2} \\ 3 & \frac{1}{2} & -\frac{1}{2} & -\frac{5}{2} \\ -1 & 0 & 0 & 1 \\ -2 & -\frac{3}{4} & \frac{1}{4} & \frac{9}{4}\end{array}\right]$


Question: A certain river is one half mile wide with a current flowing at 2 miles per hour from East to West. A man swims directly toward the opposite shore from the South bank of the river at a speed of 3 miles per hour. How far down the river does he find himself when he has swam across? How far does he end up traveling?

Answer: Water: $\left[\begin{array}{ll}-2 & 0\end{array}\right]$ Swimmer: $\left[\begin{array}{ll}0 & 3\end{array}\right]$ Speed relative to earth: $\left[\begin{array}{ll}-2 & 3\end{array}\right]$. It takes him $1 / 6$ of an hour to get across. Therefore, he ends up travelling $\frac{1}{6} \sqrt{4+9}=\frac{1}{6} \sqrt{13}$ miles. He ends up $1 / 3$ mile down stream.


Question: An object moves 10 meters in the direction of $\vec{j}$. There are two forces acting on this object, $\vec{F}_{1}=\vec{i}+\vec{j}+2 \vec{k}$, and $\vec{F}_{2}=-5 \vec{i}+2 \vec{j}-6 \vec{k}$. Find the total work done on the object by the two forces. Hint: You can take the work done by the resultant of the two forces or you can add the work done by each force. Why?

Answer:
$\left[\begin{array}{r}-4 \\ 3 \\ -4\end{array}\right] \bullet\left[\begin{array}{l}0 \\ 1 \\ 0\end{array}\right] \times 10=30$
You can consider the resultant of the two forces because of the properties of the dot product.


Question:
Find the angle between the vectors
$$
\vec{u}=\left[\begin{array}{r}
1 \\
-2 \\
1
\end{array}\right], \vec{v}=\left[\begin{array}{r}
1 \\
2 \\
-7
\end{array}\right]
$$

Answer:
$\frac{-10}{\sqrt{1+4+1} \sqrt{1+4+49}}=-0.55555=\cos \theta$
Therefore we need to solve $-0.55555=\cos \theta$, which gives $\theta=2.0313$ radians.


Question:
Find $h$ such that
$$
\left[\begin{array}{ll|l}
1 & h & 3 \\
2 & 4 & 6
\end{array}\right]
$$
is the augmented matrix of a consistent system.

Answer: Any $h$ will work.


Question: Let $A=\left[\begin{array}{ll}1 & 2 \\ 3 & 4\end{array}\right], B=\left[\begin{array}{ll}1 & 2 \\ 1 & k\end{array}\right]$. Is it possible to choose $k$ such that $A B=B A$ ? If so, what should $k$ equal?

Answer: There is no possible choice of $k$ which will make these matrices commute.


Question: Find solutions $y_{1}, y_{2}, \ldots, y_{n}$ of the equation $y^{(n)}=0$ that satisfy the initial conditions $y_{i}^{(j)}\left(x_{0}\right)=\left\{\begin{array}{ll}0, & j \neq i-1, \\1, & j=i-1,\end{array} \quad 1 \leq i \leq n .$

Answer: $y_{i}=\frac{\left(x-x_{0}\right)^{i-1}}{(i-1) !}, 1 \leq i \leq n$

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Question: Find the Wronskian of a given set $\left\{y_{1}, y_{2}\right\}$ of solutions of $(x-1) y^{\prime \prime}-x y^{\prime}+y=0$, given that $y_{1}=e^{x}$.

Answer: $y_{2}=x$

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Question: Find a curve $y=y(x)$ through $(2,1)$ such that the normal to the curve at any point $\left(x_{0}, y\left(x_{0}\right)\right)$ intersects the $y$ axis at $y_{I}=2 y\left(x_{0}\right)$.

Answer: $y=\sqrt{x^{2}-3}$

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Question: Find the general solution: $\mathbf{y}^{\prime}=\left[\begin{array}{rrr}-4 & 0 & -1 \\ -1 & -3 & -1 \\ 1 & 0 & -2\end{array}\right] \mathbf{y}$

Answer:
\mathbf{y}=\left[\begin{array}{r}
-1 \\
1 \\
0
\end{array}\right] e^{-2 t}+c_{2}\left[\begin{array}{l}
0 \\
0 \\
1
\end{array}\right] e^{-2 t}+c_{3}\left(\left[\begin{array}{r}
-1 \\
0 \\
0
\end{array}\right] e^{-2 t}+\left[\begin{array}{r}
1 \\
-1 \\
1
\end{array}\right] t e^{-2 t}\right)

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Question: Solve the equation $y^{\prime}=\frac{\cos x}{\sin y}, \quad y(\pi)=\frac{\pi}{2}$ explicitly. HINT: Use the identity $\cos (x+\pi / 2)=-\sin x$ and the periodicity of the cosine.

Answer: $y=-x+3 \pi / 2$

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Question: Find the orthogonal trajectories of the given family of curves: $x y e^{x^{2}}=c$

Answer: $y^{2}=-\frac{1}{2} \ln \left(1+2 x^{2}\right)+k$

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Question: A 3200-lb car is moving at $64 \mathrm{ft} / \mathrm{s}$ down a 30-degree grade when it runs out of fuel. Find its velocity after that if friction exerts a resistive force with magnitude proportional to the square of the speed, with $k=1 \mathrm{lb}-\mathrm{s}^{2} / \mathrm{ft}^{2}$. Also find its terminal velocity.

Answer: $v=-\frac{40\left(13+3 e^{-4 t / 5}\right)}{13-3 e^{-4 t / 5}} ;-40 \mathrm{ft} / \mathrm{s}$

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Question: Find the general solution: $\mathbf{y}^{\prime}=\left[\begin{array}{rrr}-1 & 1 & -1 \\ -2 & 0 & 2 \\ -1 & 3 & -1\end{array}\right] \mathbf{y}$

Answer:
\mathbf{y}=c_{1}\left[\begin{array}{l}
0 \\
1 \\
1
\end{array}\right] e^{2 t}+c_{2}\left[\begin{array}{l}
1 \\
0 \\
1
\end{array}\right] e^{-2 t}+c_{3}\left(\left[\begin{array}{l}
1 \\
1 \\
0
\end{array}\right] \frac{e^{-2 t}}{2}+\left[\begin{array}{l}
1 \\
0 \\
1
\end{array}\right] t e^{-2 t}\right)

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Question: Find the general solution: $\mathbf{y}^{\prime}=\left[\begin{array}{rr}3 & 2 \\ -5 & 1\end{array}\right] \mathbf{y}$

Answer: $\mathbf{y}=c_{1} e^{2 t}\left[\begin{array}{c}3 \sin 3 t-\cos 3 t \\ 5 \cos 3 t\end{array}\right]+c_{2} e^{2 t}\left[\begin{array}{c}-3 \cos 3 t-\sin 3 t \\ 5 \sin 3 t\end{array}\right]$

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Question: Find the general solution: $\mathbf{y}^{\prime}=\left[\begin{array}{rrr}1 & 2 & -2 \\ 0 & 2 & -1 \\ 1 & 0 & 0\end{array}\right] \mathbf{y}^{\prime}$

Answer: $\mathbf{y}=c_{1}\left[\begin{array}{r}6 \\ -3 \\ 3\end{array}\right] e^{8 t}+\left[\begin{array}{c}10 \cos 4 t-4 \sin 4 t \\ 17 \cos 4 t-\sin 4 t \\ 3 \cos 4 t-7 \sin 4 t\end{array}\right]$

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Question: Two objects suspended from identical springs are set into motion. The period of one object is twice the period of the other. How are the weights of the two objects related?

Answer: The object with the longer period weighs four times as much as the other.

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Question: Find the general solution of the given Euler equation on $(0, \infty)$: $3 x^{2} y^{\prime \prime}-x y^{\prime}+y=0$

Answer: $y=c_{1} x+c_{2} x^{1 / 3}$

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Question: Find the general solution of $y^{\prime \prime}+\omega_{0}^{2} y=M \cos \omega x+N \sin \omega x$, where $M$ and $N$ are constants and $\omega$ and $\omega_{0}$ are distinct positive numbers.

Answer: $y=\frac{1}{\omega_{0}^{2}-\omega^{2}}(M \cos \omega x+N \sin \omega x)+c_{1} \cos \omega_{0} x+c_{2} \sin \omega_{0} x$

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Question: Find the general solution: $y^{\prime}+a y=0(a=$ constant $)$

Answer: $y=e^{-a x}$

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Question: Find the general solution of the given Euler equation on $(0, \infty)$: $2 x^{2} y^{\prime \prime}-3 x y^{\prime}+2 y=0$

Answer: $y=c_{1} x^{2}+c_{2} x^{1 / 2}$

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Question: Find the Wronskian of a given set $\left\{y_{1}, y_{2}\right\}$ of solutions of $4 x^{2}(\sin x) y^{\prime \prime}-4 x(x \cos x+\sin x) y^{\prime}+(2 x \cos x+3 \sin x) y=0$, given that $y_{1}=x^{1 / 2}$.

Answer: $y_{2}=x^{1 / 2} \cos x$

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Question: A $2 \mathrm{lb}$ weight stretches a spring 6 inches in equilibrium. An external force $F(t)=\sin 8 t \mathrm{lb}$ is applied to the weight, which is released from rest 2 inches below equilibrium. Find its displacement for $t>0$.

Answer: $y=-t \cos 8 t-\frac{1}{6} \cos 8 t+\frac{1}{8} \sin 8 t \mathrm{ft}$

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Question: At $12: 00 \mathrm{PM}$ a thermometer reading $10^{\circ} \mathrm{F}$ is placed in a room where the temperature is $70^{\circ} \mathrm{F}$. It reads $56^{\circ}$ when it's placed outside, where the temperature is $5^{\circ} \mathrm{F}$, at $12: 03$. What does it read at 12:05 PM?

Answer: $\approx 24.33^{\circ} \mathrm{F}$

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Question: A spring with natural length $.5 \mathrm{~m}$ has length $50.5 \mathrm{~cm}$ with a mass of $2 \mathrm{gm}$ suspended from it. The mass is initially displaced $1.5 \mathrm{~cm}$ below equilibrium and released with zero velocity. Find its displacement for $t>0$.

Answer: $y=1.5 \cos 14 \sqrt{10} t \mathrm{~cm}$

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Question: An object is placed in a room where the temperature is $20^{\circ} \mathrm{C}$. The temperature of the object drops by $5^{\circ} \mathrm{C}$ in 4 minutes and by $7^{\circ} \mathrm{C}$ in 8 minutes. What was the temperature of the object when it was initially placed in the room?

Answer: $(85 / 3)^{\circ} \mathrm{C}$

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Question: Find the orthogonal trajectories of the given family of curves: $y=c e^{2 x}$

Answer: $y^{2}=-x+k$

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Question: Find a fundamental set of solutions: $\left(D^{2}+9\right)^{3} D^{2} y=0$

Answer: $\left\{\cos 3 x, x \cos 3 x, x^{2} \cos 3 x, \sin 3 x, x \sin 3 x, x^{2} \sin 3 x, 1, x\right\}$

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Question: Find the general solution: $\mathbf{y}^{\prime}=\left[\begin{array}{rr}4 & 12 \\ -3 & -8\end{array}\right] \mathbf{y}$

Answer:
c_{1}\left[\begin{array}{r}
-2 \\
1
\end{array}\right]+c_{2}\left(\left[\begin{array}{r}
-1 \\
0
\end{array}\right] \frac{e^{-2 t}}{3}+\left[\begin{array}{r}
-2 \\
1
\end{array}\right] t e^{-2 t}\right)

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Question: Find the general solution: $x^{2} y^{\prime \prime}+x y^{\prime}-y=\frac{4}{x^{2}} ; \quad y_{1}=x$

Answer: $y=\frac{4}{3 x^{2}}+c_{1} x+\frac{c_{2}}{x}$

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Question: Find a fundamental set of solutions: $(D-2)^{3}(D+1)^{2} D y=0$

Answer: $\left\{e^{2 x}, x e^{2 x}, x^{2} e^{2 x}, e^{-x}, x e^{-x}, 1\right\}$

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Question: Find the orthogonal trajectories of the given family of curves: $y=\frac{c e^{x}}{x}$

Answer: $y^{2}=-2 x-\ln (x-1)^{2}+k$

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Question: An $8 \mathrm{lb}$ weight stretches a spring $.32 \mathrm{ft}$. The weight is initially displaced 6 inches above equilibrium and given an upward velocity of $4 \mathrm{ft} / \mathrm{sec}$. Find its displacement for $t>0$ if the medium exerts a damping force of $1.5 \mathrm{lb}$ for each $\mathrm{ft} / \mathrm{sec}$ of velocity.

Answer: $y=\frac{1}{2} e^{-3 t}\left(\cos \sqrt{91} t+\frac{11}{\sqrt{91}} \sin \sqrt{91} t\right) \mathrm{ft}$

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Question: Find a power series solution $y(x)=\sum_{n=0}^{\infty} a_{n} x^{n}$ for $\left(1+3 x^{2}\right) y^{\prime \prime}-2 x y^{\prime}+4 y$.

Answer: $b_{n}=(n+2)(n+1) a_{n+2}+\left(3 n^{2}-5 n+4\right) a_{n}$

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Question: A mass of one $\mathrm{kg}$ stretches a spring $49 \mathrm{~cm}$ in equilibrium. It is attached to a dashpot that supplies a damping force of $4 \mathrm{~N}$ for each $\mathrm{m} / \mathrm{sec}$ of speed. Find the steady state component of its displacement if it's subjected to an external force $F(t)=8 \sin 2 t-6 \cos 2 t \mathrm{~N}$.

Answer: $y=-\frac{1}{2} \cos 2 t+\frac{1}{4} \sin 2 t \mathrm{~m}$

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Question:
Find the steady state current in the circuit described by the equation.
$\frac{1}{20} Q^{\prime \prime}+2 Q^{\prime}+100 Q=10 \cos 25 t-5 \sin 25 t$

Answer: $I_{p}=\frac{20}{37}(\cos 25 t-6 \sin 25 t)$

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Question: Find the general solution: $\mathbf{y}^{\prime}=\left[\begin{array}{rr}3 & 4 \\ -1 & 7\end{array}\right] \mathbf{y}$

Answer:
\mathbf{y}=c_{1}\left[\begin{array}{l}
2 \\
1
\end{array}\right] e^{5 t}+c_{2}\left(\left[\begin{array}{r}
-1 \\
0
\end{array}\right] e^{5 t}+\left[\begin{array}{l}
2 \\
1
\end{array}\right] t e^{5 t}\right)

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Question: Find the general solution: $2 x y^{\prime \prime}+(4 x+1) y^{\prime}+(2 x+1) y=3 x^{1 / 2} e^{-x} ; \quad y_{1}=e^{-x}$

Answer: $y=e^{-x}\left(x^{3 / 2}+c_{1}+c_{2} x^{1 / 2}\right)$

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Question: Find a fundamental set of solutions: $x^{2} y^{\prime \prime}-4 x y^{\prime}+6 y=0 ; \quad y_{1}=x^{2}$

Answer: $\{x^{2}, x^{3}\}$

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Question: Find the general solution: $y^{(4)}-4 y^{\prime \prime \prime}+7 y^{\prime \prime}-6 y^{\prime}+2 y=0$

Answer: $y=e^{x}\left(c_{1}+c_{2} x+c_{3} \cos x+c_{4} \sin x\right)$

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Question: Find the general solution: $y^{\prime \prime \prime}-y^{\prime \prime}-y^{\prime}+y=e^{2 x}(10+3 x)$

Answer: $y=e^{2 x}(1+x)+c_{1} e^{-x}+e^{x}\left(c_{2}+c_{3} x\right)$

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Question: Two objects suspended from identical springs are set into motion. The weight of one object is twice the weight of the other. How are the periods of the resulting motions related?

Answer: $T_{2}=\sqrt{2} T_{1}$, where $T_{1}$ is the period of the smaller object.

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Question:
Find the current in the $R L C$ circuit, assuming that $E(t)=0$ for $t>0$.
$R=2$ ohms; $L=.1$ henrys; $C=.01$ farads; $Q_{0}=2$ coulombs; $I_{0}=0$ amperes.

Answer: $I=-\frac{200}{3} e^{-10 t} \sin 30 t$

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Question: Find the general solution: $\mathbf{y}^{\prime}=\left[\begin{array}{rrr}4 & -2 & -2 \\ -2 & 3 & -1 \\ 2 & -1 & 3\end{array}\right] \mathbf{y}$

Answer:
\mathbf{y}=c_{1}\left[\begin{array}{r}
-2 \\
-3 \\
1
\end{array}\right] e^{2 t}+c_{2}\left[\begin{array}{r}
0 \\
-1 \\
1
\end{array}\right] e^{4 t}+c_{3}\left(\left[\begin{array}{l}
1 \\
0 \\
0
\end{array}\right] \frac{e^{4 t}}{2}+\left[\begin{array}{r}
0 \\
-1 \\
1
\end{array}\right] t e^{4 t}\right)

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Question: Solve the equation explicitly: $y^{\prime}=\frac{y^{2}+2 x y}{x^{2}}$

Answer: $y=\frac{c x^{2}}{1-c x} \quad y=-x$

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Question:
Find the current in the $R L C$ circuit, assuming that $E(t)=0$ for $t>0$.
$R=2$ ohms; $L=.05$ henrys; $C=.01$ farads'; $Q_{0}=2$ coulombs; $I_{0}=-2$ amperes.

Answer: $I=e^{-20 t}(2 \cos 40 t-101 \sin 40 t)$

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Question: A boat weighs $64,000 \mathrm{lb}$. Its propellor produces a constant thrust of $50,000 \mathrm{lb}$ and the water exerts a resistive force with magnitude proportional to the speed, with $k=2000 \mathrm{lb}-\mathrm{s} / \mathrm{ft}$. Assuming that the boat starts from rest, find its velocity as a function of time, and find its terminal velocity.

Answer: $v=25\left(1-e^{-t}\right) ; 25 \mathrm{ft} / \mathrm{s}$

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Question: Find the general solution: $\mathbf{y}^{\prime}=\left[\begin{array}{rrr}1 & 1 & 2 \\ 1 & 0 & -1 \\ -1 & -2 & -1\end{array}\right] \mathbf{y}$

Answer: $\mathbf{y}=c_{1}\left[\begin{array}{r}-1 \\ 1 \\ 1\end{array}\right] e^{-2 t}+c_{2} e^{t}\left[\begin{array}{r}\sin t \\ -\cos t \\ \cos t\end{array}\right]+c_{3} e^{t}\left[\begin{array}{r}-\cos t \\ -\sin t \\ \sin t\end{array}\right]$

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Question: Find the general solution of the given Euler equation on $(0, \infty)$: $2 x^{2} y^{\prime \prime}+10 x y^{\prime}+9 y=0$

Answer: $y=\frac{1}{x^{2}}\left[c_{1} \cos \left(\frac{1}{\sqrt{2}} \ln x\right)+c_{2} \sin \left(\frac{1}{\sqrt{2}} \ln x\right)\right]$

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