# Mathematics for Applied Sciences (Osnabrück 2011-2012)/Part I/Exercise sheet 8

    Warm-up-exercises

### Exercise

Let $K$ be a field, and let $V$ be a $K$-vector space of dimension $n=\dim _{\left(V\right)}$. Suppose that $n$ vectors $v_{1},\ldots ,v_{n}$ in $V$ are given. Prove that the following facts are equivalent.
* $v_{1},\ldots ,v_{n}$ form a basis for $V$.
* $v_{1},\ldots ,v_{n}$ form a system of generators for $V$.
* $v_{1},\ldots ,v_{n}$ are linearly independent.

### Exercise

Let $K$ be a field, and let $K[X]$ denote the polynomial ring over $K$. Let $d\in \mathbb {N}$. Show that the set of all polynomials of degree $\leq d$ is a finite dimensional subspace of $K[X]$. What is its dimension?

### Exercise

Show that the set of real polynomials of degree $\leq 4$ which have a zero at $-2$ and a zero at $3$ is a finite dimensional subspace of $\mathbb {R} [X]$. Determine the dimension of this vector space.

### Exercise *

Let $K$ be a field, and let $V$ and $W$ be two finite-dimensional $K$vector spaces with
    $\dim _{\left(V\right)}=n\,$
and
    $\dim _{\left(W\right)}=m\,.$
What is the dimension of the Cartesian product $V\times W$?

### Exercise

Let $V$ be a finite-dimensional vector space over the complex numbers, and let $v_{1},\ldots ,v_{n}$ be a basis of $V$. Prove that the family of vectors
    $v_{1},\ldots ,v_{n}{\text{ and }}{\mathrm {i} }v_{1},\ldots ,{\mathrm {i} }v_{n}$
form a basis for $V$, considered as a real vector space.

### Exercise

Consider the standard basis $e_{1},e_{2},e_{3},e_{4}$ in $\mathbb {R} ^{4}$ and the three vectors
    ${\begin{pmatrix}1\\3\\0\\-4\end{pmatrix}},\,{\begin{pmatrix}2\\1\\5\\7\end{pmatrix}}{\text{ and }}{\begin{pmatrix}-4\\9\\-5\\1\end{pmatrix}}.$
Prove that these vectors are linearly independent and extend them to a basis by adding an appropriate standard vector as shown in the base change theorem. Can one take any standard vector?

### Exercise

Determine the transformation matrices $M_{\mathfrak {v}}^{\mathfrak {u}}$ and $M_{\mathfrak {u}}^{\mathfrak {v}}$, for the standard basis ${\mathfrak {u}}$, and the basis ${\mathfrak {v}}$ in $\mathbb {R} ^{4}$, which is given by
    $v_{1}={\begin{pmatrix}0\\0\\1\\0\end{pmatrix}},\,v_{2}={\begin{pmatrix}1\\0\\0\\0\end{pmatrix}},\,v_{3}={\begin{pmatrix}0\\0\\0\\1\end{pmatrix}},\,v_{4}={\begin{pmatrix}0\\1\\0\\0\end{pmatrix}}.$

### Exercise

Determine the transformation matrices $M_{\mathfrak {v}}^{\mathfrak {u}}$ and $M_{\mathfrak {u}}^{\mathfrak {v}}$, for the standard basis ${\mathfrak {u}}$, and the basis ${\mathfrak {v}}$ of $\mathbb {C} ^{2}$, which is given by the vectors
    $v_{1}={\begin{pmatrix}3+5{\mathrm {i} }\\1-{\mathrm {i} }\end{pmatrix}}{\text{ and }}v_{2}={\begin{pmatrix}2+3{\mathrm {i} }\\4+{\mathrm {i} }\end{pmatrix}}.$

### Exercise

We consider the families of vectors
    ${\mathfrak {v}}={\begin{pmatrix}7\\-4\end{pmatrix}},\,{\begin{pmatrix}8\\1\end{pmatrix}}\,\,{\text{ and }}\,\,{\mathfrak {u}}={\begin{pmatrix}4\\6\end{pmatrix}},\,{\begin{pmatrix}7\\3\end{pmatrix}}$
in $\mathbb {R} ^{2}$.
a) Show that ${\mathfrak {v}}$ and ${\mathfrak {u}}$ are both a basis of $\mathbb {R} ^{2}$.
b) Let $P\in \mathbb {R} ^{2}$ denote the point which has the coordinates $(-2,5)$ with respect to the basis ${\mathfrak {v}}$. What are the coordinates of this point with respect to the basis ${\mathfrak {u}}$?
c) Determine the transformation matrix which describes the change of bases from ${\mathfrak {v}}$ to ${\mathfrak {u}}$.
    Hand-in-exercises

### Exercise (4 marks)

Show that the set of all real polynomials of degree $\leq 6$, which have a zero at $-1$, at $0$ and at $1$, is a finite dimensional subspace of $\mathbb {R} [X]$. Determine the dimension of this vector space.

### Exercise (2 marks)

Let $K$ be a field, and let $V$ be a $K$-vector space. Let $v_{1},\ldots ,v_{m}$ be a family of vectors in $V$, and let
    $U=\langle v_{i},\,i=1,\ldots ,m\rangle \,$
be the linear subspace they span. Prove that the family is linearly independent if and only if the dimension of $U$ is exactly $m$.

### Exercise (4 marks)

Determine the transformation matrices $M_{\mathfrak {v}}^{\mathfrak {u}}$ and $M_{\mathfrak {u}}^{\mathfrak {v}}$, for the standard basis ${\mathfrak {u}}$, and the basis ${\mathfrak {v}}$ of $\mathbb {R} ^{3}$, which is given by the vectors
    $v_{1}={\begin{pmatrix}4\\5\\1\end{pmatrix}},\,\,v_{2}={\begin{pmatrix}2\\3\\-8\end{pmatrix}}\,{\text{ and }}\,v_{3}={\begin{pmatrix}5\\7\\-3\end{pmatrix}}$

### Exercise (6 (3+1+2) marks)

We consider the families of vectors
    ${\mathfrak {v}}={\begin{pmatrix}1\\2\\3\end{pmatrix}},\,{\begin{pmatrix}4\\7\\1\end{pmatrix}},\,{\begin{pmatrix}0\\2\\5\end{pmatrix}}\,\,{\text{ and }}\,\,{\mathfrak {u}}={\begin{pmatrix}0\\2\\4\end{pmatrix}},\,{\begin{pmatrix}6\\6\\1\end{pmatrix}},\,{\begin{pmatrix}3\\5\\-2\end{pmatrix}}$
in $\mathbb {R} ^{3}$.
a) Show that ${\mathfrak {v}}$ and ${\mathfrak {u}}$ are both a basis of $\mathbb {R} ^{3}$.
b) Let $P\in \mathbb {R} ^{3}$ denote the point which has the coordinates $(2,5,4)$ with respect to the basis ${\mathfrak {v}}$. What are the coordinates of this point with respect to the basis ${\mathfrak {u}}$?
c) Determine the transformation matrix which describes the change of basis from ${\mathfrak {v}}$ to ${\mathfrak {u}}$.