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

Exercises

### Exercise

Write in ${\displaystyle {}\mathbb {Q} ^{2}}$ the vector

${\displaystyle (2,-7)}$

as a linear combination of the vectors

${\displaystyle (5,-3){\text{ and }}(-11,4).}$

### Exercise

Write in ${\displaystyle {}\mathbb {C} ^{2}}$ the vector

${\displaystyle (1,0)}$

as a linear combination of the vectors

${\displaystyle (3+5{\mathrm {i} },-3+2{\mathrm {i} }){\text{ and }}(1-6{\mathrm {i} },4-{\mathrm {i} }).}$

### Exercise

Let ${\displaystyle {}K}$ be a field, and let ${\displaystyle {}V}$ be a ${\displaystyle {}K}$-vector space. Show that the following statements hold.

1. For a family ${\displaystyle {}v_{i}}$, ${\displaystyle {}i\in I}$, of elements in ${\displaystyle {}V}$, linear span is a linear subspace of ${\displaystyle {}V}$.
2. The family ${\displaystyle {}v_{i}}$, ${\displaystyle {}i\in I}$, is a spanning system of ${\displaystyle {}V}$ if and only if
${\displaystyle {}\langle v_{i},\,i\in I\rangle =V\,.}$

### Exercise

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

### Exercise

Let ${\displaystyle {}K}$ be a field, and let ${\displaystyle {}V}$ be a ${\displaystyle {}K}$-vector space. Let ${\displaystyle {}v_{i}}$, ${\displaystyle {}i\in I}$, be a family of vectors in ${\displaystyle {}V}$, and let ${\displaystyle {}w\in V}$ be another vector. Assume that the family

${\displaystyle w,v_{i},i\in I,}$

is a system of generators of ${\displaystyle {}V}$, and that ${\displaystyle {}w}$ is a linear combination of the ${\displaystyle {}v_{i}}$, ${\displaystyle {}i\in I}$. Prove that also ${\displaystyle {}v_{i}}$, ${\displaystyle {}i\in I}$, is a system of generators of ${\displaystyle {}V}$.

### Exercise

We consider in ${\displaystyle {}\mathbb {Q} ^{3}}$ the linear subspaces

${\displaystyle {}U=\langle {\begin{pmatrix}2\\1\\4\end{pmatrix}},{\begin{pmatrix}3\\-2\\7\end{pmatrix}}\rangle \,}$

and

${\displaystyle {}W=\langle {\begin{pmatrix}5\\-1\\11\end{pmatrix}},{\begin{pmatrix}1\\-3\\3\end{pmatrix}}\rangle \,.}$

Show that ${\displaystyle {}U=W}$.

### Exercise

Show that the three vectors

${\displaystyle {\begin{pmatrix}0\\1\\2\\1\end{pmatrix}},\,{\begin{pmatrix}4\\3\\0\\2\end{pmatrix}},\,{\begin{pmatrix}1\\7\\0\\-1\end{pmatrix}}}$

in ${\displaystyle {}\mathbb {R} ^{4}}$ are linearly independent.

### Exercise

Find, for the vectors

${\displaystyle {\begin{pmatrix}7\\-5\\3\end{pmatrix}},\,{\begin{pmatrix}-4\\1\\-6\end{pmatrix}},\,{\begin{pmatrix}2\\8\\0\end{pmatrix}},\,{\begin{pmatrix}5\\-5\\8\end{pmatrix}}}$

in ${\displaystyle {}\mathbb {Q} ^{3}}$, a non-trivial representation of the zero vector.

### Exercise

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

### Exercise

Let ${\displaystyle {}K}$ be a field, let ${\displaystyle {}V}$ be a ${\displaystyle {}K}$-vector space and let ${\displaystyle {}v_{i}}$, ${\displaystyle {}i\in I}$, be a family of vectors in ${\displaystyle {}V}$. Prove the following facts.

1. If the family is linearly independent, then for each subset ${\displaystyle {}J\subseteq I}$, also the family ${\displaystyle {}v_{i}}$ , ${\displaystyle {}i\in J}$ is linearly independent.
2. The empty family is linearly independent.
3. If the family contains the null vector, then it is not linearly independent.
4. If a vector appears several times in the family, then the family is not linearly independent.
5. A vector ${\displaystyle {}v}$ is linearly independent if and only if ${\displaystyle {}v\neq 0}$.
6. Two vectors ${\displaystyle {}v}$ and ${\displaystyle {}u}$ are linearly independent if and only if ${\displaystyle {}u}$ is not a scalar multiple of ${\displaystyle {}v}$ and vice versa.

### Exercise

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

### Exercise

Determine a basis for the solution space of the linear equation

${\displaystyle {}3x+4y-2z+5w=0\,.}$

### Exercise

Determine a basis for the solution space of the linear system of equations

${\displaystyle -2x+3y-z+4w=0{\text{ and }}3z-2w=0.}$

### Exercise

Prove that in ${\displaystyle {}\mathbb {R} ^{3}}$, the three vectors

${\displaystyle {\begin{pmatrix}2\\1\\5\end{pmatrix}}\,,{\begin{pmatrix}1\\3\\7\end{pmatrix}}\,,{\begin{pmatrix}4\\1\\2\end{pmatrix}}}$

are a basis.

### Exercise

Establish if in ${\displaystyle {}\mathbb {C} ^{2}}$ the two vectors

${\displaystyle {\begin{pmatrix}2+7{\mathrm {i} }\\3-{\mathrm {i} }\end{pmatrix}}{\text{ and }}{\begin{pmatrix}15+26{\mathrm {i} }\\13-7{\mathrm {i} }\end{pmatrix}}}$

form a basis.

### Exercise

Let ${\displaystyle {}K}$ be a field. Find a linear system of equations in three variables, whose solution space is exactly

${\displaystyle {\left\{\lambda {\begin{pmatrix}3\\2\\-5\end{pmatrix}}\mid \lambda \in K\right\}}.}$

### Exercise

Let ${\displaystyle {}K}$ be a field, and let

${\displaystyle {}{\begin{pmatrix}a_{1}\\\vdots \\a_{n}\end{pmatrix}}\in K^{n}\,}$

be a nonzero vector. Find a linear system of equations in ${\displaystyle {}n}$ variables with ${\displaystyle {}n-1}$ equations, whose solution space is exactly

${\displaystyle {\left\{\lambda {\begin{pmatrix}a_{1}\\\vdots \\a_{n}\end{pmatrix}}\mid \lambda \in K\right\}}.}$

### Exercise

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

1. ${\displaystyle {}v_{1},\ldots ,v_{n}}$ form a basis for ${\displaystyle {}V}$.
2. ${\displaystyle {}v_{1},\ldots ,v_{n}}$ form a system of generators for ${\displaystyle {}V}$.
3. ${\displaystyle {}v_{1},\ldots ,v_{n}}$ are linearly independent.

### Exercise

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

### Exercise

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

### Exercise *

Let ${\displaystyle {}K}$ be a field, and let ${\displaystyle {}V}$ and ${\displaystyle {}W}$ be two finite-dimensional ${\displaystyle {}K}$vector spaces with

${\displaystyle {}\dim _{}{\left(V\right)}=n\,}$

and

${\displaystyle {}\dim _{}{\left(W\right)}=m\,.}$

What is the dimension of the Cartesian product ${\displaystyle {}V\times W}$?

### Exercise

Let ${\displaystyle {}V}$ be a finite-dimensional vector space over the complex numbers, and let ${\displaystyle {}v_{1},\ldots ,v_{n}}$ be a basis of ${\displaystyle {}V}$. Prove that the family of vectors

${\displaystyle v_{1},\ldots ,v_{n}{\text{ and }}{\mathrm {i} }v_{1},\ldots ,{\mathrm {i} }v_{n}}$

form a basis for ${\displaystyle {}V}$, considered as a real vector space.

### Exercise

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

Hand-in-exercises

### Exercise (3 marks)

Write in ${\displaystyle {}\mathbb {Q} ^{3}}$ the vector

${\displaystyle (2,5,-3)}$

as a linear combination of the vectors

${\displaystyle (1,2,3),(0,1,1){\text{ und }}(-1,2,4).}$

Prove that it cannot be expressed as a linear combination of two of the three vectors.

### Exercise (4 marks)

We consider in ${\displaystyle {}\mathbb {Q} ^{4}}$ the linear subspaces

${\displaystyle {}U=\langle {\begin{pmatrix}3\\1\\-5\\2\end{pmatrix}},{\begin{pmatrix}2\\-2\\4\\-3\end{pmatrix}},{\begin{pmatrix}1\\0\\3\\2\end{pmatrix}}\rangle \,}$

and

${\displaystyle {}W=\langle {\begin{pmatrix}6\\-1\\2\\1\end{pmatrix}},{\begin{pmatrix}0\\-2\\-2\\-7\end{pmatrix}},{\begin{pmatrix}9\\2\\-1\\10\end{pmatrix}}\rangle \,.}$

Show that ${\displaystyle {}U=W}$.

### Exercise (2 marks)

Establish if in ${\displaystyle {}\mathbb {R} ^{3}}$ the three vectors

${\displaystyle {\begin{pmatrix}2\\3\\-5\end{pmatrix}}\,,{\begin{pmatrix}9\\2\\6\end{pmatrix}}\,,{\begin{pmatrix}-1\\4\\-1\end{pmatrix}}}$

form a basis.

### Exercise (2 marks)

Establish if in ${\displaystyle {}\mathbb {C} ^{2}}$ the two vectors

${\displaystyle {\begin{pmatrix}2-7{\mathrm {i} }\\-3+2{\mathrm {i} }\end{pmatrix}}{\text{ and }}{\begin{pmatrix}5+6{\mathrm {i} }\\3-17{\mathrm {i} }\end{pmatrix}}}$

form a basis.

### Exercise (4 marks)

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

### Exercise (4 marks)

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

### Exercise (2 marks)

Let ${\displaystyle {}K}$ be a field, and let ${\displaystyle {}V}$ be a ${\displaystyle {}K}$-vector space. Let ${\displaystyle {}v_{1},\ldots ,v_{m}}$ be a family of vectors in ${\displaystyle {}V}$, and let

${\displaystyle {}U=\langle v_{i},\,i=1,\ldots ,m\rangle \,}$

be the linear subspace they span. Prove that the family is linearly independent if and only if the dimension of ${\displaystyle {}U}$ is exactly ${\displaystyle {}m}$.