# Mathematics for Applied Sciences (Osnabrück 2023-2024)/Part I/Exercise sheet 23

    Exercises

### Exercise

Write in $\mathbb {Q} ^{2}$ the vector
    $(2,-7)$
as a linear combination of the vectors
    $(5,-3){\text{ and }}(-11,4).$

### Exercise

Write in $\mathbb {C} ^{2}$ the vector
    $(1,0)$
as a linear combination of the vectors
    $(3+5{\mathrm {i} },-3+2{\mathrm {i} }){\text{ and }}(1-6{\mathrm {i} },4-{\mathrm {i} }).$

### Exercise

Let $K$ be a field, and let $V$ be a $K$-vector space. Show that the following statements hold.
* For a family $v_{i}$, $i\in I$, of elements in $V$, linear span is a linear subspace of $V$.
* The family $v_{i}$, $i\in I$, is a spanning system of $V$ if and only if
    $\langle v_{i},\,i\in I\rangle =V\,.$

### Exercise

Let $K$ be a field, and let $V$ be a $K$-vector space. Let $v_{i}$, $i\in I$, be a family of vectors in $V$ and $w_{j}$, $j\in J$, another family of vectors in $V$. Then, for the spanned linear subspaces, the inclusion $\langle v_{i},\,i\in I\rangle \subseteq \langle w_{j},\,j\in J\rangle$ holds, if and only if $v_{i}\in \langle w_{j},\,j\in J\rangle$ holds for all $i\in I$.

### Exercise

Let $K$ be a field, and let $V$ be a $K$-vector space. Let $v_{i}$, $i\in I$, be a family of vectors in $V$, and let $w\in V$ be another vector. Assume that the family
    $w,v_{i},i\in I,$
is a system of generators of $V$, and that $w$ is a linear combination of the $v_{i}$, $i\in I$. Prove that also $v_{i}$, $i\in I$, is a system of generators of $V$.

### Exercise

We consider in $\mathbb {Q} ^{3}$ the linear subspaces
    $U=\langle {\begin{pmatrix}2\\1\\4\end{pmatrix}},{\begin{pmatrix}3\\-2\\7\end{pmatrix}}\rangle \,$
and
    $W=\langle {\begin{pmatrix}5\\-1\\11\end{pmatrix}},{\begin{pmatrix}1\\-3\\3\end{pmatrix}}\rangle \,.$
Show that $U=W$.

### Exercise

Show that the three vectors
    ${\begin{pmatrix}0\\1\\2\\1\end{pmatrix}},\,{\begin{pmatrix}4\\3\\0\\2\end{pmatrix}},\,{\begin{pmatrix}1\\7\\0\\-1\end{pmatrix}}$
in $\mathbb {R} ^{4}$ are linearly independent.

### Exercise

Find, for the vectors
    ${\begin{pmatrix}7\\-5\\3\end{pmatrix}},\,{\begin{pmatrix}-4\\1\\-6\end{pmatrix}},\,{\begin{pmatrix}2\\8\\0\end{pmatrix}},\,{\begin{pmatrix}5\\-5\\8\end{pmatrix}}$
in $\mathbb {Q} ^{3}$, a non-trivial representation of the zero-vector.

### Exercise

Give an example of three vectors in $\mathbb {R} ^{3}$ such that each two of them is linearly independent, but all three vectors together are linearly dependent.

### Exercise

Let $K$ be a field, let $V$ be a $K$-vector space and let $v_{i}$, $i\in I$, be a family of vectors in $V$. Prove the following facts.
* If the family is linearly independent, then for each subset $J\subseteq I$, also the family $v_{i}$ , $i\in J$ is linearly independent.
* The empty family is linearly independent.
* If the family contains the null vector, then it is not linearly independent.
* If a vector appears several times in the family, then the family is not linearly independent.
* A vector $v$ is linearly independent if and only if $v\neq 0$.
* Two vectors $v$ and $u$ are linearly independent if and only if $u$ is not a scalar multiple of $v$ and vice versa.

### Exercise

Let $K$ be a field, let $V$ be a $K$-vector space, and let $v_{i}$, $i\in I$ be a family of vectors in $V$. Let $\lambda _{i}$, $i\in I$ be a family of elements $\neq 0$ in $K$. Prove that the family $v_{i}$, $i\in I$, is linearly independent (a system of generators of $V$, a basis of $V$), if and only if the same holds for the family $\lambda _{i}v_{i}$, $i\in I$.

### Exercise

Determine a basis for the solution space of the linear equation
    $3x+4y-2z+5w=0\,.$

### Exercise

Determine a basis for the solution space of the linear system of equations
    $-2x+3y-z+4w=0{\text{ and }}3z-2w=0.$

### Exercise

Prove that in $\mathbb {R} ^{3}$, the three vectors
    ${\begin{pmatrix}2\\1\\5\end{pmatrix}}\,,{\begin{pmatrix}1\\3\\7\end{pmatrix}}\,,{\begin{pmatrix}4\\1\\2\end{pmatrix}}$
are a basis.

### Exercise

Establish if in $\mathbb {C} ^{2}$ the two vectors
    ${\begin{pmatrix}2+7{\mathrm {i} }\\3-{\mathrm {i} }\end{pmatrix}}{\text{ and }}{\begin{pmatrix}15+26{\mathrm {i} }\\13-7{\mathrm {i} }\end{pmatrix}}$
form a basis.

### Exercise

Let $K$ be a field. Find a linear system of equations in three variables, whose solution space is exactly
    ${\left\{\lambda {\begin{pmatrix}3\\2\\-5\end{pmatrix}}\mid \lambda \in K\right\}}.$

### Exercise

Let $K$ be a field, and let
    ${\begin{pmatrix}a_{1}\\\vdots \\a_{n}\end{pmatrix}}\in K^{n}\,$
be a nonzero vector. Find a linear system of equations in $n$ variables with $n-1$ equations, whose solution space is exactly
    ${\left\{\lambda {\begin{pmatrix}a_{1}\\\vdots \\a_{n}\end{pmatrix}}\mid \lambda \in K\right\}}.$

### 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 all real polynomials of degree $\leq 4$, which have a zero for $-2$ and for $3$, form a finite-dimensional linear subspace in $\mathbb {R} [X]$. Determine its dimension.

### 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

Let $K$ be a finite field with $q$ elements, and let $V$ be an $n$-dimensional vector space. Let $v_{1},v_{2},v_{3},\ldots$ be an enumeration (without repetitions) of the elements from $V$. After how many elements can we be sure that these form a generating system of $V$.
    Hand-in-exercises

### Exercise (3 marks)

Write in $\mathbb {Q} ^{3}$ the vector
    $(2,5,-3)$
as a linear combination of the vectors
    $(1,2,3),(0,1,1){\text{ und }}(-1,2,4).$
Prove that it cannot be expressed as a linear combination of two of the three vectors.

### Exercise (4 marks)

We consider in $\mathbb {Q} ^{4}$ the linear subspaces
    $U=\langle {\begin{pmatrix}3\\1\\-5\\2\end{pmatrix}},{\begin{pmatrix}2\\-2\\4\\-3\end{pmatrix}},{\begin{pmatrix}1\\0\\3\\2\end{pmatrix}}\rangle \,$
and
    $W=\langle {\begin{pmatrix}6\\-1\\2\\1\end{pmatrix}},{\begin{pmatrix}0\\-2\\-2\\-7\end{pmatrix}},{\begin{pmatrix}9\\2\\-1\\10\end{pmatrix}}\rangle \,.$
Show that $U=W$.

### Exercise (2 marks)

Establish if in $\mathbb {R} ^{3}$ the three vectors
    ${\begin{pmatrix}2\\3\\-5\end{pmatrix}}\,,{\begin{pmatrix}9\\2\\6\end{pmatrix}}\,,{\begin{pmatrix}-1\\4\\-1\end{pmatrix}}$
form a basis.

### Exercise (2 marks)

Establish if in $\mathbb {C} ^{2}$ the two vectors
    ${\begin{pmatrix}2-7{\mathrm {i} }\\-3+2{\mathrm {i} }\end{pmatrix}}{\text{ and }}{\begin{pmatrix}5+6{\mathrm {i} }\\3-17{\mathrm {i} }\end{pmatrix}}$
form a basis.

### Exercise (4 marks)

Let $\mathbb {Q} ^{n}$ be the $n$-dimensional standard vector space over $\mathbb {Q}$, and let $v_{1},\ldots ,v_{n}\in \mathbb {Q} ^{n}$ be a family of vectors. Prove that this family is a $\mathbb {Q}$-basis of $\mathbb {Q} ^{n}$ if and only if the same family, considered as a family in $\mathbb {R} ^{n}$, is a $\mathbb {R}$-basis of $\mathbb {R} ^{n}$.

### 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 $m$.