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

Exercise for the break

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Show by an example of two basesMDLD/bases (vs) $v,u$ and $v,w$ in ${ℝ}^2$, that the coordinate functionMDLD/coordinate function (vs) $v^{\ast }$ depend on the basis, and not only on $v$.

Exercises

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Let

$U=⟨\left(\begin{array}{c}4\\ 6\\ 7\end{array}\right),\left(\begin{array}{c}3\\ -3\\ 8\end{array}\right)⟩\subseteq {ℝ}^3.$

Find a linear formMDLD/linear form $f:{ℝ}^3\to ℝ$ such that $U=\mathrm{kern}f$ holds.

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Solve the linear systemMDLD/linear system

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

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Show that the real partMDLD/real part and the imaginary partMDLD/imaginary part define real linear formsMDLD/linear forms on $ℂ$, where $ℂ$ is considered as a real vector space.

Is the modulusMDLD/modulus (C) of a complex number a real linear form?

===Exercise * Exercise 14.5

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Let $V$ be an $n$-dimensionalMDLD/dimensional (vs) $K$-vector space,MDLD/vector space and let $U\subseteq V$ denote an $n-1$-dimensional linear subspace.MDLD/linear subspace Show that there exists a linear formMDLD/linear form $f:V\to K$ such that $U=\mathrm{kern}f$.

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Let $K$ denote a field,MDLD/field let $V$ be a $K$-vector space,MDLD/vector space and $U\subseteq V$ a linear subspace.MDLD/linear subspace Let $v\in V$ with $v\notin U$. Show that there exists a linear formMDLD/linear form $\phi :V\to K$ satisfying $\phi (U)=0$ and $\phi (v)=1$.

===Exercise Exercise 14.7

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Let $K$ be a field,MDLD/field and let $V$ be a $K$-vector space.MDLD/vector space Let $v_1,\dots ,v_n\in V$ be vectors. Suppose that for every $k$, there exists a linear formMDLD/linear form

${\phi }_k:V⟶K$

such that

${\phi }_k(v_k)\ne 0\text{ and }{\phi }_k(v_i)=0\text{ for }i\ne k.$

Show that the $v_1,\dots ,v_n$ are linearly independent.MDLD/linearly independent

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Let $V$ be a finite-dimensionalMDLD/finite-dimensional (vs) real vector space.MDLD/real vector space Show that a linear mappingMDLD/linear mapping

$f:V⟶ℝ,$

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

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Let $V$ be a finite-dimensionalMDLD/finite-dimensional (vs) $K$-vector spaceMDLD/vector space over a fieldMDLD/field $K$, and let $L,L_1,\dots ,L_m$ denote linear formsMDLD/linear forms on $V$. Show that the relation

$\bigcap _{i=1}^m\mathrm{kern}{L}_i\subseteq \mathrm{kern}L$

holds if and only if $L$ belongs to the linear subspaceMDLD/linear subspace (in the dual spaceMDLD/dual space) generatedMDLD/generated (vs) by the $L_1,\dots ,L_m$.

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Express the vectors $u_1^{\ast },u_2^{\ast }$ of the dual basisMDLD/dual basis of the basis $u_1=\left(\begin{array}{c}1\\ 3\end{array}\right),u_2=\left(\begin{array}{c}2\\ -5\end{array}\right)$ in ${ℝ}^2$ as linear combinationsMDLD/linear combinations with respect to the standard dual basis $e_1^{\ast },e_2^{\ast }$.

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Express the vectors $e_1^{\ast },e_2^{\ast }$ of the standard dual basisMDLD/standard dual basis as linear combinationsMDLD/linear combinations with respect to the dual basisMDLD/dual basis $u_1^{\ast },u_2^{\ast }$ to the basisMDLD/basis (vs) $u_1=\left(\begin{array}{c}1\\ 4\end{array}\right),u_2=\left(\begin{array}{c}-2\\ 2\end{array}\right)$.

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Let $V$ and $W$ be vector spacesMDLD/vector spaces over a fieldMDLD/field $K$, with a basisMDLD/basis (vs) $v_1,\dots ,v_n$ of $V$, and a basis $w_1,\dots ,w_m$ of $W$. Show that

$v_i^{\ast }{w}_j,i=1,\dots ,n,j=1,\dots ,m,$

is a basis of the space of homomorphismsMDLD/space of homomorphisms ${\mathrm{Hom}}_K(V,W)$.

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Let $V$ be a $K$-vector space,MDLD/vector space together with its dual spaceMDLD/dual space $V^{\ast }$. Show that the natural mapping

$V\times V^{\ast }⟶K,(v,f)⟼f(v),$

is not linear.MDLD/linear

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

$\mathrm{trace}(A\circ B)=\mathrm{trace}(B\circ A).$

===Exercise Exercise 14.15

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

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Let $K$ be a field,MDLD/field and let $V$ be a finite-dimensionalMDLD/finite-dimensional (vs) $K$-vector space.MDLD/vector space Show that the assignment

${\mathrm{End}}_{}\left(V\right)⟶K,\phi ⟼\mathrm{trace}\left(\phi \right),$

is $K$-linear.MDLD/linear

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Determine the traceMDLD/trace (linear) of a linear projectionMDLD/linear projection

$\phi :V⟶V$

on a finite-dimensionalMDLD/finite-dimensional $K$-vector spaceMDLD/vector space $V$.

Hand-in-exercises

Exercise (3 marks) Create referencenumber

Let

$U=⟨\left(\begin{array}{c}9\\ 2\\ -9\end{array}\right),\left(\begin{array}{c}13\\ 23\\ 33\end{array}\right)⟩\subseteq {ℝ}^3.$

Find a linear formMDLD/linear form $f:{ℝ}^3\to ℝ$ such that $U=\mathrm{kern}f$.

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

Let $K$ be a fieldMDLD/field and $a,b,c\in K$.

a) Show that the vectors

$\left(\begin{array}{c}b\\ -a\\ 0\end{array}\right),\left(\begin{array}{c}0\\ c\\ -b\end{array}\right),\left(\begin{array}{c}c\\ 0\\ -a\end{array}\right)\in K^3$

are solutions of the linear equation

$ax+by+cz=0.$

b) Show that these three vectors are linearly independent.MDLD/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) Create referencenumber

Express the vectors $u_1^{\ast },u_2^{\ast }$ of the dual basisMDLD/dual basis of the basis $u_1=\left(\begin{array}{c}4\\ 7\end{array}\right),u_2=\left(\begin{array}{c}6\\ -1\end{array}\right)$ in ${ℝ}^2$ as linear combinationsMDLD/linear combinations with respect to the standard dual basis $e_1^{\ast },e_2^{\ast }$.

Exercise (4 marks) Create referencenumber

Express the vectors $u_1^{\ast },u_2^{\ast },u_3^{\ast }$ of the dual basisMDLD/dual basis of the basis $u_1=\left(\begin{array}{c}4\\ -2\\ 1\end{array}\right),u_2=\left(\begin{array}{c}5\\ 3\\ 2\end{array}\right),u_3=\left(\begin{array}{c}0\\ -1\\ 7\end{array}\right)$ in ${ℝ}^3$ as linear combinationsMDLD/linear combinations with respect to the standard dual basis $e_1^{\ast },e_2^{\ast },e_3^{\ast }$.

Exercise (2 marks) Create referencenumber

Let $V={\mathrm{Mat}}_n(K)$ be the space of the $n\times n$-matricesMDLD/matrices over the field $K$, with the standard basis $e_{ij}$. Describe the traceMDLD/trace (linear) as a linear combinationMDLD/linear combination with respect to the dual basisMDLD/dual basis $e_{ij}^{\ast }$.

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