Linear algebra (Osnabrück 2024-2025)/Part II/Exercise sheet 56

Exercises

Exercise

We consider in ${}\mathbb {R} ^{2}$ the bases ${}{\mathfrak {v}}={\begin{pmatrix}3\\-7\end{pmatrix}},{\begin{pmatrix}-5\\2\end{pmatrix}}$ and the standard basis ${}{\mathfrak {e}}$, and we consider in ${}\mathbb {C}$ the real bases ${}{\mathfrak {x}}=8-2{\mathrm {i} },6+3{\mathrm {i} }$ and ${}{\mathfrak {y}}=1,{\mathrm {i} }$. Determine the transformation matrix of the corresponding bases of the tensor product ${}\mathbb {R} ^{2}\otimes _{\mathbb {R} }\mathbb {C}$.

Exercise

Let ${}K$ be a field, and let ${}U,V,W$ denote ${}K$-vector spaces. Show that the following statements hold. (in the sense of a canonical isomorphy).

Exercise

The linear mappings

$\varphi \colon \mathbb {R} ^{2}\longrightarrow \mathbb {R} ^{2}$

and

$\psi \colon \mathbb {R} ^{2}\longrightarrow \mathbb {R} ^{2}$

are given with respect to the standard bases by the matrices ${}{\begin{pmatrix}3&2\\-4&9\end{pmatrix}}$ and ${}{\begin{pmatrix}6&3\\0&5\end{pmatrix}}$. Determine the matrix of the linear mapping

$\varphi \otimes \psi \colon \mathbb {R} ^{2}\otimes \mathbb {R} ^{2}\longrightarrow \mathbb {R} ^{2}\otimes \mathbb {R} ^{2}.$

Exercise

Let ${}K$ be a field, and let ${}Z$ be a vector space over ${}K$. We consider the assignment ${}V\mapsto V\otimes _{K}Z$ that maps a vector space ${}V$ tothe tensor product ${}V\otimes _{K}Z$, and a ${}K$-linear mapping

$\varphi \colon V\longrightarrow W$

to the tensorization ${}\varphi \otimes \operatorname {Id} _{Z}$. Show the following statements.

1. For the identity

   $\operatorname {Id} _{V}\colon V\longrightarrow V,$

   also

   ${}\operatorname {Id} _{V}\otimes \operatorname {Id} _{Z}=\operatorname {Id} _{V\otimes Z}\,$

   is the identity.

2. For a linear mapping

   $U{\stackrel {\varphi }{\longrightarrow }}V{\stackrel {\psi }{\longrightarrow }}W$

   we have

   ${}(\psi \circ \varphi )\otimes \operatorname {Id} _{Z}={\left(\varphi \otimes \operatorname {Id} _{Z}\right)}\circ {\left(\psi \otimes \operatorname {Id} _{Z}\right)}\,.$

3. For an isomorphism

   $\varphi \colon V\longrightarrow W,$

   also ${}\varphi \otimes \operatorname {Id} _{Z}$ is an isomorphism, and for the inverse mapping we have

   ${}{\left(\varphi \otimes \operatorname {Id} _{Z}\right)}^{-1}=\varphi ^{-1}\otimes \operatorname {Id} _{Z}\,.$

Exercise

Let ${}K$ be a field, and let ${}U,V,W$ denote ${}K$-vector spaces. Show that we have

${}U\otimes _{K}{\left(V\oplus W\right)}\cong {\left(U\otimes _{K}V\right)}\oplus {\left(U\otimes _{K}W\right)}\,.$

Exercise

Let ${}V_{1},\ldots ,V_{n}$ be finite-dimensional ${}K$-vector spaces, and let

$U_{1,1}\subset U_{1,2}\subset \ldots \subset U_{1,\dim _{K}{\left(V_{1}\right)}}\subset V_{1},\ldots ,U_{n,1}\subset U_{n,2}\subset \ldots \subset U_{n,\dim _{K}{\left(V_{n}\right)}}\subset V_{n}$

be flags in these vector spaces. Show that, in general, there does not exist a flag in ${}V_{1}\otimes _{K}\cdots \otimes _{K}V_{n}$ such that the linear subspaces involved have the form

$U_{1,j_{1}}\otimes _{}\cdots \otimes _{}U_{n,j_{n}}.$

Exercise

Let ${}U_{1}\subseteq V_{1},U_{2}\subseteq V_{2},\ldots ,U_{n}\subseteq V_{n}$ be linear subspaces with the residue class spaces ${}V_{1}/U_{1},\ldots ,V_{n}/U_{n}$. Does there exist a canonical isomorphism

${}(V_{1}\otimes _{}\cdots \otimes _{}V_{n})/(U_{1}\otimes _{}\cdots \otimes _{}U_{n})\cong (V_{1}/U_{1})\otimes _{}\cdots \otimes _{}(V_{n}/U_{n})\,?$

Exercise

Let ${}K$ be a field, and let ${}V_{1},\ldots ,V_{n}$ be vector spaces over ${}K$. Let diagonalizable ${}K$-linear mappings

$\varphi _{i}\colon V_{i}\longrightarrow V_{i}$

be given. Show that also the tensor product

$\varphi _{1}\otimes _{}\cdots \otimes _{}\varphi _{n}\colon V_{1}\otimes _{}\cdots \otimes _{}V_{n}\longrightarrow V_{1}\otimes _{}\cdots \otimes _{}V_{n}$

is diagonalizable.

Exercise

Let ${}K$ be a field, and let ${}V_{1},\ldots ,V_{n}$ be vector spaces over ${}K$. Let trigonalizable ${}K$-linear mappings

$\varphi _{i}\colon V_{i}\longrightarrow V_{i}$

be given. Show that also the tensor product

$\varphi _{1}\otimes _{}\cdots \otimes _{}\varphi _{n}\colon V_{1}\otimes _{}\cdots \otimes _{}V_{n}\longrightarrow V_{1}\otimes _{}\cdots \otimes _{}V_{n}$

is trigonalizable.

The Kronecker product of matrices

${}A={\left(a_{ij}\right)}_{1\leq i\leq m,\,1\leq j\leq n}\,$

and

${}B={\left(b_{k\ell }\right)}_{1\leq k\leq p,\,1\leq \ell \leq r}\,$

is given by

${\left(a_{ij}\cdot b_{k\ell }\right)}_{1\leq i\leq m,\,1\leq k\leq p;\,1\leq j\leq n,\,1\leq \ell \leq r}.$

Exercise

Compute the Kronecker product of the two matrices ${}{\begin{pmatrix}3&-4\\5&-2\end{pmatrix}}$ and ${}{\begin{pmatrix}-2&7\\6&3\end{pmatrix}}$.

The Kronecker product has a tight connection with the tensor product, as the following exercise shows.

Exercise

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

${}A={\left(a_{ij}\right)}_{1\leq i\leq m,\,1\leq j\leq n}\,$

and

${}B={\left(b_{k\ell }\right)}_{1\leq k\leq p,\,1\leq \ell \leq r}\,$

denote matrices with the corresponding linear mappings ${}A\colon K^{n}\rightarrow K^{m}$ and ${}B\colon K^{r}\rightarrow K^{p}$. Show that the tensor product of these linear mappings is given by the Kroneckerprodukt of ${}A$ and ${}B$ with respect to the bases ${}e_{j}\otimes e_{\ell }$, ${}1\leq j\leq n,\,1\leq \ell \leq r$, of ${}K^{n}\otimes K^{r}$ and ${}e_{i}\otimes e_{k}$, ${}1\leq i\leq m,\,1\leq k\leq p$, of ${}K^{m}\otimes K^{p}$.

Exercise

Suppose that the linear mapping ${}\varphi \colon V\rightarrow V$ is given by the Jordan matrix ${}{\begin{pmatrix}1&1\\0&1\end{pmatrix}}$ with respect to the basis ${}v_{1},v_{2}$, and that the linear mapping ${}\psi \colon W\rightarrow W$ is given by the Jordan matrix ${}{\begin{pmatrix}1&1\\0&1\end{pmatrix}}$ with respect to the basis ${}w_{1},w_{2}$.

1. Determine the matrix of

   $\varphi \otimes \psi \colon V\otimes W\longrightarrow V\otimes W$

   with respect to the basis ${}v_{1}\otimes w_{1},v_{1}\otimes w_{2},v_{2}\otimes w_{1},v_{2}\otimes w_{2}$.

2. Determine the Jordan normal form of ${}\varphi \otimes \psi$.

Exercise

Let ${}K$ be a field, and let ${}V_{1},\ldots ,V_{n},W_{1},\ldots ,W_{n}$ be vector spaces over ${}K$. Show that the mapping

$\operatorname {Hom} _{K}{\left(V_{1},W_{1}\right)}\times \cdots \times \operatorname {Hom} _{K}{\left(V_{n},W_{n}\right)}\longrightarrow \operatorname {Hom} _{K}{\left(V_{1}\otimes _{}\cdots \otimes _{}V_{n},W_{1}\otimes _{}\cdots \otimes _{}W_{n}\right)},(\varphi _{1},\ldots ,\varphi _{n})\longmapsto \varphi _{1}\otimes _{}\cdots \otimes _{}\varphi _{n},$

is multilinear.

Exercise

Let ${}K$ be a field, and let ${}V_{1},\ldots ,V_{n},W_{1},\ldots ,W_{n}$ denote finite-dimensional ${}K$-vector spaces. Show that there exists a canonical isomorphism

$\operatorname {Hom} _{K}{\left(V_{1},W_{1}\right)}\otimes _{}\cdots \otimes _{}\operatorname {Hom} _{K}{\left(V_{n},W_{n}\right)}\longrightarrow \operatorname {Hom} _{K}{\left(V_{1}\otimes _{}\cdots \otimes _{}V_{n},W_{1}\otimes _{}\cdots \otimes _{}W_{n}\right)}.$

Exercise

Let ${}K\subseteq L$ be a field extension, and let ${}f\in L$. Show that the mapping

$\mu _{f}\colon L\longrightarrow L,x\longmapsto fx,$

is ${}K$-linear.

Exercise

Let ${}K\subseteq L$ be a field extension, and let ${}a\in L$. Show that the inserting mapping, that is, the assignment

$\psi \colon K[X]\longrightarrow L,P\longmapsto P(a),$

fulfills the following properties (for ${}P,Q\in K[X]$).

1. ${}(P+Q)(a)=P(a)+Q(a)$,
2. ${}(P\cdot Q)(a)=P(a)\cdot Q(a)$,
3. ${}1(a)=1$.

Exercise

Let ${}K$ be a field, ${}V$ be a ${}K$-vector space, and ${}K\subseteq L$ be a field extension. Let ${}v_{i}$, ${}i\in I$, be a family of vectors in ${}V$. Show the following statements.

1. The family ${}v_{i}$, ${}i\in I$, is a ${}K$-generating system of ${}V$ if and only if ${}1\otimes v_{i}$, ${}i\in I$, is an ${}L$-generating system of ${}L\otimes V$.
2. The family ${}v_{i}$, ${}i\in I$, is ${}K$-linearly independent in ${}V$ if and only if ${}1\otimes v_{i}$, ${}i\in I$, is linearly independent (over ${}L$) in ${}L\otimes V$.
3. Die family ${}v_{i}$, ${}i\in I$, is a ${}K$-basis of ${}V$ if and only if ${}1\otimes v_{i}$, ${}i\in I$, is an ${}L$-basis of ${}L\otimes V$.

Exercise

Let ${}K\subseteq L$ be a field extension. We consider the assignment ${}V\mapsto V_{L}=L\otimes _{K}V$, that maps a ${}K$-vector space ${}V$ to the ${}L$-vector space ${}L\otimes _{K}V$, and a ${}K$-linear mapping

$\varphi \colon V\longrightarrow W$

to the tensorization ${}\varphi _{L}$. Show the following statements.

1. For the identity

   $\operatorname {Id} _{V}\colon V\longrightarrow V,$

   also

   ${}{\left(\operatorname {Id} _{V}\right)}_{L}=\operatorname {Id} _{L\otimes _{K}V}\,$

   is the identity.

2. For linear mappings

   $U{\stackrel {\varphi }{\longrightarrow }}V{\stackrel {\psi }{\longrightarrow }}W$

   we have

   ${}(\psi \circ \varphi )_{L}=\psi _{L}\circ \varphi _{L}\,.$

3. For an isomorphism

   $\varphi \colon V\longrightarrow W,$

   also ${}\varphi _{L}$ is an isomorphism, and for the inverse mapping we have

   ${}{\left(\varphi _{L}\right)}^{-1}={\left(\varphi ^{-1}\right)}_{L}\,.$

A field extension ${}K\subseteq L$ is called finite if ${}L$ is a finite-dimensional vector space

over ${}K$.

Let ${}K\subseteq L$ be a finite field extension. Then the ${}K$-vector space dimension

of ${}L$ is called the degree of the field extension.

Exercise

Determine the degree of the field extension ${}\mathbb {R} \subseteq \mathbb {C}$.

Exercise

Let ${}K\subseteq L\subseteq M$ be field extensions such that ${}M$ is finite over ${}K$. Show that also ${}M$ is finite over ${}L$ and that ${}L$ is finite over ${}K$.

Exercise

Let ${}K\subseteq L$ be a finite field extension, and let ${}x_{1},\ldots ,x_{n}\in L$ denote a ${}K$-basis of ${}L$. Show that the multiplication on ${}L$ is uniquely determined by the products

$x_{i}x_{j},1\leq i\leq j\leq n.$

Exercise

Let ${}K\subseteq L$ be a finite field extension, and let ${}v_{1},\ldots ,v_{n}\in L$ denote elements that form a ${}K$-basis of ${}L$. Let ${}x\in L$, ${}x\neq 0$. Show that also ${}xv_{1},\ldots ,xv_{n}\in L$ form a ${}K$-basis of ${}L$.

Exercise

Let ${}K\subseteq L$ be a field extension, and let ${}V_{1},\ldots ,V_{n}$ denote ${}K$-vector spaces. Show that there exists a canonical isomorphism of ${}L$-vector spaces

${}L\otimes _{K}{\left(V_{1}\otimes _{K}\cdots \otimes _{K}V_{n}\right)}={\left(L\otimes _{K}V_{1}\right)}\otimes _{L}\cdots \otimes _{L}{\left(L\otimes _{K}V_{n}\right)}\,.$

Exercise

Show that the field extension ${}\mathbb {Q} \subseteq \mathbb {R}$ is not finite.

Let ${}K$ be a field, and let ${}A$ denote a ${}K$-vector space. ${}A$ is called a commutative ${}K$-Algebra if there exists a fixed element ${}1$ and a binary operation, called multiplication,

$A\times A\longrightarrow A,(a,b)\longmapsto a\cdot b,$

such that the following conditions hold.

1. We have

   ${}1\cdot a=a\,$

   for all ${}a\in A$.

2. The operation is associative.
3. We have

   ${}a\cdot b=b\cdot a\,$

   for all ${}a\in A$.

4. For ${}c\in K$ and ${}a\in A$, we have

   ${}ca=(c1)\cdot a\,,$

   where ${}ca$ and ${}c1$ means the scalar multiplication.

Important examples of ${}K$-algebras are given by a field extension ${}K\subseteq L$. The polynomial ring ${}K[X]$ is a ${}K$-algebra as well.

Exercise

Let ${}K$ be a field, and let ${}A$ and ${}B$ denote algebras over ${}K$. Show that ${}A\otimes _{K}B$ is also a ${}K$-algebra, where the ${}1$ is given by ${}1\otimes 1$ and where the multiplication for decomposable tensors are given by

${}(a\otimes b)\cdot (c\otimes d):=(a\cdot c)\otimes (b\cdot d)\,.$

In den following exercises, the symbol ${}\cong$ means that these is a correspondence that respects addition, the multiplication, the ${}0$, and the ${}1$.

Exercise

Let ${}K\subseteq L$ be a field extension. Show that for the polynomial ring, the equality

${}L\otimes _{K}K[X]\cong L[X]\,$

holds.

Exercise

Let ${}K$ be a field. Show that for polynomial rings, the equality

${}K[X]\otimes _{K}K[Y]\cong K[X,Y]\,$

holds.

Exercise

Let ${}K\subseteq L$ be a field extension, and let ${}A$ be a ${}K$-algebra. Show that ${}L\otimes _{K}A$ is an ${}L$-algebra.

Hand-in-exercises

Exercise (4 marks)

We consider in ${}\mathbb {R} ^{2}$ the bases ${}{\mathfrak {v}}={\begin{pmatrix}6\\-5\end{pmatrix}},{\begin{pmatrix}-4\\7\end{pmatrix}}$ and the standard basis ${}{\mathfrak {e}}$, and we consider in ${}\mathbb {R} ^{3}$ the basis ${}{\mathfrak {x}}={\begin{pmatrix}-3\\4\\-5\end{pmatrix}},{\begin{pmatrix}2\\-1\\6\end{pmatrix}},{\begin{pmatrix}5\\2\\4\end{pmatrix}}$ and the standard basis. Determine the transformation matrix of the corresponding bases of the tensor product ${}\mathbb {R} ^{2}\otimes _{\mathbb {R} }\mathbb {R} ^{3}$.

Exercise (2 marks)

Let ${}K$ be a field, and let ${}U_{1},\ldots ,U_{n},V_{1},\ldots ,V_{n},W_{1},\ldots ,W_{n}$ denote vector spaces over ${}K$. Let

$\psi _{i}\colon U_{i}\longrightarrow V_{i}$

and

$\varphi _{i}\colon V_{i}\longrightarrow W_{i}$

be ${}K$-linear mappings. Show

${}(\varphi _{1}\circ \psi _{1})\otimes _{}\cdots \otimes _{}(\varphi _{n}\circ \psi _{n})=(\varphi _{1}\otimes _{}\cdots \otimes _{}\varphi _{n})\circ (\psi _{1}\otimes _{}\cdots \otimes _{}\psi _{n})\,.$

Exercise (2 marks)

Let ${}V_{1},\ldots ,V_{n}$ be vector spaces over the field ${}K$, and let

$\varphi _{i}\colon V_{i}\longrightarrow V_{i}$

denote linear mappings, and let

$\varphi =\varphi _{1}\otimes _{}\cdots \otimes _{}\varphi _{n}\colon V_{1}\otimes _{}\cdots \otimes _{}V_{n}\longrightarrow V_{1}\otimes _{}\cdots \otimes _{}V_{n}$

be the corresponding tensor product mapping. Let ${}a_{i}$ be an eigenvalue of ${}\varphi _{i}$. Show that ${}a_{1}\cdots a_{n}$ is an eigenvalue of ${}\varphi$.

Exercise (4 marks)

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

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

that arises from the identification

${}\operatorname {End} _{}{\left(V\right)}={V}^{*}\otimes _{K}V\,$

according to Exercise 55.15 and from the natural mapping

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

in the sense of Exercise 55.14 , equals the trace.

Exercise (6 (2+4) marks)

Suppose that the linear mapping ${}\varphi \colon V\rightarrow V$ is given by the Jordan matrix ${}{\begin{pmatrix}1&1\\0&1\end{pmatrix}}$ with respect to the basis ${}v_{1},v_{2}$, and that the linear mapping ${}\psi \colon W\rightarrow W$ is given by the Jordan matrix ${}{\begin{pmatrix}1&1&0\\0&1&1\\0&0&1\end{pmatrix}}$ with respect to the basis ${}w_{1},w_{2},w_{3}$.

1. Determine the matrix of

   $\varphi \otimes \psi \colon V\otimes W\longrightarrow V\otimes W$

   with respect to the basis ${}v_{1}\otimes w_{1},v_{1}\otimes w_{2},v_{1}\otimes w_{3},v_{2}\otimes w_{1},v_{2}\otimes w_{2},v_{2}\otimes w_{3}$.

2. Determine the Jordan normal form of ${}\varphi \otimes \psi$.

Exercise (4 marks)

Let ${}U_{1}\subseteq V_{1},U_{2}\subseteq V_{2},\ldots ,U_{n}\subseteq V_{n}$ be linear subspaces with the residue class spaces ${}V_{1}/U_{1},\ldots ,V_{n}/U_{n}$. Show that the kernel of the canonical mapping

$V_{1}\otimes _{}\cdots \otimes _{}V_{n}\longrightarrow (V_{1}/U_{1})\otimes _{}\cdots \otimes _{}(V_{n}/U_{n})$

is

$\sum _{j}V_{1}\otimes _{}\cdots \otimes _{}V_{j-1}\otimes U_{j}\otimes V_{j+1}\otimes _{}\cdots \otimes _{}V_{n}.$

Exercise (4 marks)

Let ${}K\subseteq L$ be a field extension. Show that ${}L\otimes _{K}L$ is not necessarily a field.

<< | Linear algebra (Osnabrück 2024-2025)/Part II | >> PDF-version of this exercise sheet Lecture for this exercise sheet (PDF)