Linear algebra (Osnabrück 2024-2025)/Part II/Exercise sheet 55

Exercises

We first recall the following five exercises, the tensor product sheds new light on them.

Exercise

Let ${}K$ be a field, and ${}I$ an index set. Show that

{}K^{I}:=\operatorname {Maps} \,(I,K)\,,

with pointwise addition and scalar multiplication, is a ${}K$ -vector space.

Exercise

Let ${}K$ be a field, let ${}I$ denote an index set, and let ${}K^{I}=\operatorname {Map} \,{\left(I,K\right)}$ be the corresponding vector space.

a) Show that

{}E={\left\{f\in K^{I}\mid f(i)=0{\text{ for all }}i\in I{\text{ up to finite exceptions}}\right\}}\,

is a linear subspace of ${}K^{I}$ .

b) For every ${}i\in I$ , let ${}e_{i}\in K^{I}$ be defined by

{}e_{i}(j)={\begin{cases}1,{\text{ if }}j=i\,,\\0{\text{ else}}\,.\end{cases}}\,

Show that every element ${}f\in E$ can be expressed uniquely as a linear combination of the family ${}e_{i}$ , ${}i\in I$ .

Exercise

Let ${}K$ be a field, and let ${}S$ and ${}T$ be sets. Show that a mapping

\psi \colon S\longrightarrow T

defines a linear mapping

\operatorname {Map} \,{\left(T,K\right)}\longrightarrow \operatorname {Map} \,{\left(S,K\right)},\varphi \longmapsto \varphi \circ \psi .

Exercise

Let ${}K$ be a field, and let ${}S$ and ${}T$ denote sets. Let

\psi \colon S\longrightarrow T

be a mapping.

a) Show that, by ${}e_{s}\mapsto e_{\psi (s)}$ , a linear mapping

K^{(S)}\longrightarrow K^{(T)}

is determined.

b) Suppose now that ${}\psi$ has also the property that all its fibers are finite. Show that this defines a linear mapping

K^{(T)}\longrightarrow K^{(S)},\varphi \longmapsto \varphi \circ \psi .

Exercise

Show that, in general, in a tensor product ${}V\otimes _{K}W$ , not every vector is of the form ${}v\otimes w$ .

In the following exercises, the word compute means to express the tensor product as a linear combination of the tensor products of the standard vectors.

Exercise

Compute in ${}\mathbb {R} ^{2}\otimes _{\mathbb {R} }\mathbb {R} ^{2}$ the tensor product

{\begin{pmatrix}6\\8\end{pmatrix}}\otimes {\begin{pmatrix}5\\-2\end{pmatrix}}.

Exercise

Compute in ${}\mathbb {R} ^{2}\otimes _{\mathbb {R} }\mathbb {R} ^{3}$ the tensor product

{\begin{pmatrix}-7\\3\end{pmatrix}}\otimes {\begin{pmatrix}3\\-2\\4\end{pmatrix}}.

Exercise

Compute in ${}\mathbb {R} ^{3}\otimes _{\mathbb {R} }\mathbb {R} ^{3}$ the tensor product

{\begin{pmatrix}5\\3\\7\end{pmatrix}}\otimes {\begin{pmatrix}-8\\9\\-4\end{pmatrix}}.

Exercise

Compute in ${}\mathbb {R} ^{2}\otimes _{\mathbb {R} }\mathbb {R} ^{3}\otimes _{\mathbb {R} }\mathbb {R} ^{2}$ the tensor product

{\begin{pmatrix}6\\-7\end{pmatrix}}\otimes {\begin{pmatrix}-7\\3\\-3\end{pmatrix}}\otimes {\begin{pmatrix}-5\\4\end{pmatrix}}.

Exercise

Compute in ${}\mathbb {R} ^{2}\otimes _{\mathbb {R} }\mathbb {R} ^{3}\otimes _{\mathbb {R} }\mathbb {R} ^{4}$ the tensor product

{\begin{pmatrix}2\\-{\frac {3}{7}}\end{pmatrix}}\otimes {\begin{pmatrix}-7\\{\frac {5}{4}}\\{\frac {3}{5}}\end{pmatrix}}\otimes {\begin{pmatrix}3\\1\\0\\{\frac {4}{3}}\end{pmatrix}}.

Exercise

Compute in ${}\mathbb {C} ^{2}\otimes _{\mathbb {C} }\mathbb {C} ^{3}$ the tensor product

{\begin{pmatrix}3-5{\mathrm {i} }\\4+{\mathrm {i} }\end{pmatrix}}\otimes {\begin{pmatrix}6\\1-{\mathrm {i} }\\2+3{\mathrm {i} }\end{pmatrix}}.

Exercise

Compute in ${}\mathbb {C} \otimes _{\mathbb {R} }\mathbb {C} \otimes _{\mathbb {R} }\mathbb {C}$ the tensor product

(4-5{\mathrm {i} })\otimes (3-7{\mathrm {i} })\otimes (-2-6{\mathrm {i} }).

Exercise

Let ${}K$ be a field, and let ${}V$ and ${}W$ denote a ${}K$ -vector space. Let ${}{V}^{*}$ be the dual space of ${}V$ . Show the following statements.

a) There exists a multilinear mapping

{V}^{*}\times W\longrightarrow \operatorname {Hom} _{K}{\left(V,W\right)},(f,w)\longmapsto {\left(v\mapsto f(v)w\right)}.

b) There exists a linear mapping

\psi \colon {V}^{*}\otimes W\longrightarrow \operatorname {Hom} _{K}{\left(V,W\right)}

that sends ${}f\otimes w$ to the linear mapping ${}v\mapsto f(v)w$ .

c) If ${}V$ and ${}W$ are finite-dimensional, then ${}\psi$ from part (b) is an isomorphism.

Let ${}K$ be a field, and let ${}(V_{1},\left\langle -,-\right\rangle _{1}),\ldots ,(V_{n},\left\langle -,-\right\rangle _{n})$ denote vector spaces over ${}K$ , each endowed with a fixed bilinear form ${}\left\langle -,-\right\rangle _{i}$ . Show that on the tensor product ${}V_{1}\otimes _{}\cdots \otimes _{}V_{n}$ , there exists a bilinear form ${}\left\langle -,-\right\rangle$ that satisfies

{}\left\langle v_{1}\otimes _{}\cdots \otimes _{}v_{n},w_{1}\otimes _{}\cdots \otimes _{}w_{n}\right\rangle =\left\langle v_{1},w_{1}\right\rangle _{1}\cdots \left\langle v_{n},w_{n}\right\rangle _{n}\,.

Exercise

Suppose that the ${}\mathbb {R} ^{4}$ is endowed with the Minkowski standard form. Determine the corresponding linear form on ${}\mathbb {R} ^{4}\otimes _{\mathbb {R} }\mathbb {R} ^{4}$ .

Hand-in-exercises

Exercise (2 marks)

Compute in ${}\mathbb {R} ^{3}\otimes _{\mathbb {R} }\mathbb {R} ^{3}$ the tensor product

{\begin{pmatrix}6\\-3\\8\end{pmatrix}}\otimes {\begin{pmatrix}-5\\-1\\4\end{pmatrix}}.

Exercise (3 marks)

Compute in ${}\mathbb {C} ^{2}\otimes _{\mathbb {C} }\mathbb {C} ^{3}\otimes _{\mathbb {C} }\mathbb {C} ^{2}$ the tensor product

{\begin{pmatrix}2-{\mathrm {i} }\\-{\mathrm {i} }\end{pmatrix}}\otimes {\begin{pmatrix}5\\3+2{\mathrm {i} }\\4-3{\mathrm {i} }\end{pmatrix}}\otimes {\begin{pmatrix}-1+{\mathrm {i} }\\2+{\mathrm {i} }\end{pmatrix}}.

Exercise (3 marks)

Let ${}V$ be a finite-dimensional ${}K$ -vector space, and let ${}\left\langle -,-\right\rangle$ denote a bilinear form on ${}V$ . Suppose that this form is described with respect to the basis ${}v_{1},\ldots ,v_{d}$ of ${}V$ by the Gram matrix