Linear algebra (Osnabrück 2024-2025)/Part II/Exercise sheet 57

Exercises

Exercise

Let ${}K\subseteq L$ be a field extension, ${}V$ be a finite-dimensional ${}K$-vector space, and let

$\varphi \colon V\longrightarrow V$

be a linear mapping. Show that the characteristic polynomial of ${}\varphi$ coincides with the characteristic polynomial of the tensorization ${}\varphi _{L}$.

However, by going from ${}K$ to ${}L$, there may arise new zeroes of the characteristic polynomial and hence also new eigenvalues and eigenvectors.

Exercise

Let ${}K\subseteq L$ be a field extension, and let ${}V$ and ${}W$ be vector spaces over ${}K$.

a) Define an ${}L$-linear mapping

$L\otimes _{K}\operatorname {Hom} _{K}{\left(V,W\right)}\longrightarrow \operatorname {Hom} _{L}{\left(V_{L},W_{L}\right)}$

that maps ${}c\otimes \varphi$ to ${}c{\left(\operatorname {Id} _{L}\otimes \varphi \right)}$.

b) Suppose that the two vector spaces are finite-dimensional. Show that the mapping from part (a) is an isomorphism.

Exercise

Let ${}K\subseteq L$ be a field extension, let ${}V$ be a ${}K$-vector space and ${}W$ an ${}L$-vector space. Let

$\varphi \colon V\longrightarrow W$

be a ${}K$-linear mapping. Show that there exists an ${}L$-linear mapping

${\tilde {\varphi }}\colon V_{L}\longrightarrow W$

that extends ${}\varphi$ (that is, coincides on ${}V\subseteq V_{L}$ with ${}\varphi$.).

Exercise

Simplify in ${}\bigwedge ^{2}\mathbb {R} ^{3}$ the expression

$e_{1}\wedge (e_{2}-4e_{3})-e_{2}\wedge (5e_{1}+3e_{2}-4e_{3})+{\left(7e_{3}\right)}\wedge e_{1}-4{\left(5e_{1}\wedge 2e_{3}\right)}+(2e_{1}-8e_{2})\wedge (2e_{1}-8e_{2}).$

Exercise

Simplify in ${}\bigwedge ^{3}\mathbb {R} ^{3}$ the expression

$(e_{1}-e_{2})\wedge (e_{2}-4e_{3})\wedge (3e_{1}-5e_{2}+4e_{3})+{\left(7e_{3}\right)}\wedge {\left(e_{1}-4e_{3}\right)}\wedge {\left(5e_{3}-e_{1}\right)}-(4e_{3}-2e_{2})\wedge e_{2}\wedge (4e_{1}+3e_{2}-4e_{3}).$

Exercise

Simplify in ${}\bigwedge ^{3}\mathbb {R} ^{3}$ the expression

$-7e_{1}\wedge e_{2}\wedge e_{3}+6e_{2}\wedge e_{3}\wedge e_{1}+4e_{3}\wedge e_{2}\wedge e_{1}+3e_{2}\wedge e_{1}\wedge e_{3}+5e_{3}\wedge e_{1}\wedge e_{2}-11e_{2}\wedge e_{3}\wedge e_{1}.$

Exercise

Let ${}K$ be a field, and let ${}V$ be a ${}K$-vector space. Show the equality ${}V=\bigwedge ^{1}V$.

Exercise

Let ${}K$ be a field, and let ${}V$ be a ${}K$-vector space of dimension ${}n$. Show that ${}\bigwedge ^{n}V$ is not the zero space.

Exercise

Let ${}K$ be a field, and let ${}V$ be an ${}m$-dimensional ${}K$-vector space. Let ${}n>m$. Show ${}\bigwedge ^{n}V=0$.

Exercise

Let ${}K$ denote a field, and let ${}V$ denote a finite-dimensional vector space. Let ${}f_{1},\ldots ,f_{k}\in V^{*}$. Show that the mapping

$V\times \cdots \times V\longrightarrow K,(v_{1},\ldots ,v_{k})\longmapsto \det(f_{i}(v_{j}))_{1\leq i,j\leq k},$

is multilinear and alternating.

Exercise

Prove Theorem 57.10 directly from the construction of the tensor product and the construction of the wedge product.

Exercise

Let ${}V$ be a ${}K$-vector space, and ${}n\in \mathbb {N}$.

1. Can we define by the assignment

   $v_{1}\wedge \ldots \wedge v_{n}\mapsto v_{1}\otimes _{}\cdots \otimes _{}v_{n}$

   a (linear) mapping from ${}\bigwedge ^{n}V$ to ${}V\otimes _{}\cdots \otimes _{}V$?

2. Can we apply to the canonical mapping

   $\pi \colon V\times \cdots \times V\longrightarrow V\otimes _{}\cdots \otimes _{}V$

   the universal property of the wedge product, in order to obtain a linear mapping from ${}\bigwedge ^{n}V$ to ${}V\otimes _{}\cdots \otimes _{}V$?

Exercise

Let ${}V$ be a ${}K$-vector space, and let

$\pi \colon V\times \cdots \times V\longrightarrow V\otimes _{}\cdots \otimes _{}V$

(${}n$ factors) the canonical multilinear mapping.

1. Let ${}\sigma \in S_{n}$ be a permutation. Show that there exists a multilinear mapping

   $\pi \circ \sigma \colon V\times \cdots \times V\longrightarrow V\otimes _{}\cdots \otimes _{}V$

   with

   ${}(\pi \circ \sigma )(v_{1},\ldots ,v_{n})=v_{\sigma (1)}\otimes _{}\cdots \otimes _{}v_{\sigma (n)}\,.$

2. Show that ${}\sum _{\sigma \in S_{n}}\operatorname {sgn} (\sigma )\pi \circ \sigma$ is multilinear and alternating.
3. Show that there exists a linear mapping

   $\Psi \colon \bigwedge ^{n}V\longrightarrow V\otimes _{}\cdots \otimes _{}V$

   with

   ${}\Psi (v_{1}\wedge \ldots \wedge v_{n})=\sum _{\sigma \in S_{n}}\operatorname {sgn} (\sigma )v_{\sigma (1)}\otimes _{}\cdots \otimes _{}v_{\sigma (n)}\,.$

Exercise

Let ${}K\subseteq L$ be a field extension, let ${}V$ be a ${}K$-vector space, and ${}n\in \mathbb {N}$. Show that there exists a canonical isomorphy of ${}L$-vector spaces

$\bigwedge ^{n}V_{L}\,\,{\text{ and }}\,\,(\bigwedge ^{n}V)_{L}$

(where on the left-hand side, we have the wedge product over ${}L$).

Hand-in-exercises

Exercise (2 marks)

Let ${}K\subseteq L$ be a field extension, let ${}V$ be a finite-dimensional ${}K$-vector space, and let

$\varphi \colon V\longrightarrow V$

denote a linear mapping. Show

${}\det \varphi =\det \varphi _{L}\,.$

Exercise (4 marks)

Let

$\varphi \colon V\longrightarrow V$

be an endomorphism on a finite-dimensional real vector space ${}V$, and let

$\varphi _{\mathbb {C} }\colon V_{\mathbb {C} }\longrightarrow V_{\mathbb {C} }$

be the corresponding complexification. Show that ${}\varphi$ is (asymptotically) stable if and only if this holds for ${}\varphi _{\mathbb {C} }$.

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