Linear algebra (Osnabrück 2024-2025)/Part I/Exercise sheet 14

Exercise for the break

Exercise

Show by an example of two bases ${}v,u$ and ${}v,w$ in ${}\mathbb {R} ^{2}$, that the coordinate function ${}v^{*}$ depend on the basis, and not only on ${}v$.

Exercises

Exercise

Let

${}U=\langle {\begin{pmatrix}4\\6\\7\end{pmatrix}},\,{\begin{pmatrix}3\\-3\\8\end{pmatrix}}\rangle \subseteq \mathbb {R} ^{3}\,.$

Find a linear form ${}f\colon \mathbb {R} ^{3}\rightarrow \mathbb {R}$ such that ${}U=\operatorname {kern} f$ holds.

Exercise

Solve the linear system

${}4x+7y-3z+6u+5v=0\,.$

Exercise

Show that the real part and the imaginary part define real linear forms on ${}\mathbb {C}$, where ${}\mathbb {C}$ is considered as a real vector space.

Is the modulus of a complex number a real linear form?

Exercise *

Let ${}V$ be an ${}n$-dimensional ${}K$-vector space, and let ${}U\subseteq V$ denote an ${}n-1$-dimensional linear subspace. Show that there exists a linear form ${}f\colon V\rightarrow K$ such that ${}U=\operatorname {kern} f$.

Exercise

Let ${}K$ denote a field, let ${}V$ be a ${}K$-vector space, and ${}U\subseteq V$ a linear subspace. Let ${}v\in V$ with ${}v\notin U$. Show that there exists a linear form ${}\varphi \colon V\rightarrow K$ satisfying ${}\varphi (U)=0$ and ${}\varphi (v)=1$.

Exercise

Let ${}K$ be a field, and let ${}V$ be a ${}K$-vector space. Let ${}v_{1},\ldots ,v_{n}\in V$ be vectors. Suppose that for every ${}k$, there exists a linear form

$\varphi _{k}\colon V\longrightarrow K$

such that

$\varphi _{k}(v_{k})\neq 0{\text{ and }}\varphi _{k}(v_{i})=0{\text{ for }}i\neq k.$

Show that the ${}v_{1},\ldots ,v_{n}$ are linearly independent.

Exercise

Let ${}V$ be a finite-dimensional real vector space. Show that a linear mapping

$f\colon V\longrightarrow \mathbb {R} ,$

different from ${}0$, does not have a local extrema. Does this also hold for infinite-dimensional vector spaces? Does this require analysis?

Exercise

Let ${}V$ be a finite-dimensional ${}K$-vector space over a field ${}K$, and let ${}L,L_{1},\ldots ,L_{m}$ denote linear forms on ${}V$. Show that the relation

${}\bigcap _{i=1}^{m}\operatorname {kern} L_{i}\subseteq \operatorname {kern} L\,$

holds if and only if ${}L$ belongs to the linear subspace (in the dual space) generated by the ${}L_{1},\ldots ,L_{m}$.

Exercise

Express the vectors ${}u_{1}^{*},u_{2}^{*}$ of the dual basis of the basis ${}u_{1}={\begin{pmatrix}1\\3\end{pmatrix}},\,u_{2}={\begin{pmatrix}2\\-5\end{pmatrix}}$ in ${}\mathbb {R} ^{2}$ as linear combinations with respect to the standard dual basis ${}e_{1}^{*},e_{2}^{*}$.

Exercise

Express the vectors ${}e_{1}^{*},e_{2}^{*}$ of the standard dual basis as linear combinations with respect to the dual basis ${}u_{1}^{*},u_{2}^{*}$ to the basis ${}u_{1}={\begin{pmatrix}1\\4\end{pmatrix}}\,,u_{2}={\begin{pmatrix}-2\\2\end{pmatrix}}$.

Exercise

Let ${}V$ and ${}W$ be vector spaces over a field ${}K$, with a basis ${}v_{1},\ldots ,v_{n}$ of ${}V$, and a basis ${}w_{1},\ldots ,w_{m}$ of ${}W$. Show that

$v_{i}^{*}w_{j},i=1,\ldots ,n,j=1,\ldots ,m,$

is a basis of the space of homomorphisms ${}\operatorname {Hom} _{K}{\left(V,W\right)}$.

Exercise

Let ${}V$ be a ${}K$-vector space, together with its dual space ${}{V}^{*}$. Show that the natural mapping

$V\times {V}^{*}\longrightarrow K,(v,f)\longmapsto f(v),$

is not linear.

Exercise

Let ${}K$ be a field, and let ${}A$ denote an ${}m\times n$-matrix and let ${}B$ denote an ${}n\times m$-matrix over ${}K$. Show

${}\operatorname {trace} {\left(A\circ B\right)}=\operatorname {trace} {\left(B\circ A\right)}\,.$

Exercise

Show that the definition of the trace of a linear mapping is independent of the chosen matrix.

Exercise

Let ${}K$ be a field, and let ${}V$ be a finite-dimensional ${}K$-vector space. Show that the assignment

$\operatorname {End} _{}{\left(V\right)}\longrightarrow K,\varphi \longmapsto \operatorname {trace} {\left(\varphi \right)},$

is ${}K$-linear.

Exercise

Determine the trace of a linear projection

$\varphi \colon V\longrightarrow V$

on a finite-dimensional ${}K$-vector space ${}V$.

Hand-in-exercises

Exercise (3 marks)

Let

${}U=\langle {\begin{pmatrix}9\\2\\-9\end{pmatrix}},\,{\begin{pmatrix}13\\23\\33\end{pmatrix}}\rangle \subseteq \mathbb {R} ^{3}\,.$

Find a linear form ${}f\colon \mathbb {R} ^{3}\rightarrow \mathbb {R}$ such that ${}U=\operatorname {kern} f$.

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

Let ${}K$ be a field and ${}a,b,c\in K$.

a) Show that the vectors

${}{\begin{pmatrix}b\\-a\\0\end{pmatrix}},\,{\begin{pmatrix}0\\c\\-b\end{pmatrix}},\,{\begin{pmatrix}c\\0\\-a\end{pmatrix}}\in K^{3}\,$

are solutions of the linear equation

${}ax+by+cz=0\,.$

b) Show that these three vectors are linearly independent.

c) Under what conditions generate these vectors the solution space of the equation?

d) Under what conditions generate the first two vectors the solution space of the equation?

Exercise (3 marks)

Express the vectors ${}u_{1}^{*},u_{2}^{*}$ of the dual basis of the basis ${}u_{1}={\begin{pmatrix}4\\7\end{pmatrix}},\,u_{2}={\begin{pmatrix}6\\-1\end{pmatrix}}$ in ${}\mathbb {R} ^{2}$ as linear combinations with respect to the standard dual basis ${}e_{1}^{*},e_{2}^{*}$.

Exercise (4 marks)

Express the vectors ${}u_{1}^{*},u_{2}^{*},u_{3}^{*}$ of the dual basis of the basis ${}u_{1}={\begin{pmatrix}4\\-2\\1\end{pmatrix}},\,u_{2}={\begin{pmatrix}5\\3\\2\end{pmatrix}},\,u_{3}={\begin{pmatrix}0\\-1\\7\end{pmatrix}}$ in ${}\mathbb {R} ^{3}$ as linear combinations with respect to the standard dual basis ${}e_{1}^{*},e_{2}^{*},e_{3}^{*}$.

Exercise (2 marks)

Let ${}V=\operatorname {Mat} _{n}(K)$ be the space of the ${}n\times n$-matrices over the field ${}K$, with the standard basis ${}e_{ij}$. Describe the trace as a linear combination with respect to the dual basis ${}e_{ij}^{*}$.

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