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.

1. For a family ${}v_{i}$, ${}i\in I$, of elements in ${}V$, linear span is a linear subspace of ${}V$.
2. 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.

1. If the family is linearly independent, then for each subset ${}J\subseteq I$, also the family ${}v_{i}$ , ${}i\in J$ 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 ${}v$ is linearly independent if and only if ${}v\neq 0$.
6. 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

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

form 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 ${}n=\dim _{K}{\left(V\right)}$. Suppose that ${}n$ vectors ${}v_{1},\ldots ,v_{n}$ in ${}V$ are given. Prove that the following facts are equivalent.

1. ${}v_{1},\ldots ,v_{n}$ form a basis for ${}V$.
2. ${}v_{1},\ldots ,v_{n}$ form a system of generators for ${}V$.
3. ${}v_{1},\ldots ,v_{n}$ 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 linear subspace of ${}K[X]$. What is its dimension?

Exercise

Show that the set of all real polynomials of degree ${}\leq 4$ that have a zero for ${}-2$ and for ${}3$, forms 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

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

and

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

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

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

forms 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

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

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 an ${}\mathbb {R}$-basis of ${}\mathbb {R} ^{n}$.

Exercise (4 marks)

Show that the set of all real polynomials of degree ${}\leq 6$ that 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 ${}m$.

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