Linear algebra (Osnabrück 2024-2025)/Part II/Exercise sheet 58

Exercises

Exercise

Express the wedge product ${}{\begin{pmatrix}2\\3\\2\end{pmatrix}}\wedge {\begin{pmatrix}4\\-1\\5\end{pmatrix}}$ in the standard basis of ${}\bigwedge ^{2}\mathbb {R} ^{3}$.

Exercise

Let

$\varphi \colon \mathbb {R} ^{2}\longrightarrow \mathbb {R} ^{3}$

be the linear mapping given by the matrix

${\begin{pmatrix}4&-1\\0&7\\2&3\end{pmatrix}}.$

Determine the matrix of ${}\bigwedge ^{2}\varphi$ with respect to the standard bases of the wedge products.

Exercise

Let ${}V$ be a finite-dimensional ${}K$-vector space, and let ${}v_{1},\ldots ,v_{m}$ be a basis of ${}V$. Determine the matrix of the natural mapping (${}n$ factors)

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

with respect to the corresponding bases.

Exercise

Let ${}V=U\oplus W$ be a direct sum decomposition in linear subspaces of dimension ${}\dim _{K}{\left(U\right)}=r$ and ${}\dim _{K}{\left(U\right)}=s$. Show that there exists a canonical isomorphy

${}\bigwedge ^{r+s}V\cong \bigwedge ^{r}U\otimes \bigwedge ^{s}W\,.$

Exercise

Let ${}K$ be a field, and let ${}V$ denote a ${}K$-vector space. Let ${}u_{1},\ldots ,u_{n}\in V$. Show that for every ${}k\in \mathbb {N}$, there exists a uniquely determined linear mapping

$\bigwedge ^{k}V\longrightarrow \bigwedge ^{k+n}V$

satisfying ${}v_{1}\wedge \ldots \wedge v_{k}\mapsto v_{1}\wedge \ldots \wedge v_{k}\wedge u_{1}\wedge \ldots \wedge u_{n}$.

Exercise

Let ${}V$ be a vector space over the field ${}K$, and let

$\varphi \colon V\longrightarrow V$

be an endomorphism. Let

$\bigwedge ^{m}\varphi \colon \bigwedge ^{m}V\longrightarrow \bigwedge ^{m}V$

be the ${}m$-th wedge product of ${}\varphi$. Let ${}v_{1},\ldots ,v_{m}$ be linearly independent eigenvectors of ${}\varphi$ with eigenvalues ${}a_{1},\ldots ,a_{m}$. Show that ${}a_{1}\cdots a_{m}$ is an eigenvalue of ${}\bigwedge ^{m}\varphi$.

Exercise

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

$\varphi \colon V\longrightarrow V$

be a diagonalizable ${}K$-linear mapping. Show that also the wedge product

$\bigwedge ^{n}\varphi \colon \bigwedge ^{n}V\longrightarrow \bigwedge ^{n}V$

is diagonalizable.

Exercise

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

$\varphi \colon V\longrightarrow V$

be a trigonalizable ${}K$-linear mapping. Show that also the wedge product

$\bigwedge ^{n}\varphi \colon \bigwedge ^{n}V\longrightarrow \bigwedge ^{n}V$

is trigonalizable.

Exercise

Prove the multiplication theorem for the determinant with the help of the wedge product.

Hand-in-exercises

Exercise (4 marks)

Express the wedge product

$4{\begin{pmatrix}2\\3\\4\end{pmatrix}}\wedge {\begin{pmatrix}1\\0\\3\end{pmatrix}}+5{\begin{pmatrix}-3\\2\\3\end{pmatrix}}\wedge {\begin{pmatrix}0\\0\\1\end{pmatrix}}-2{\begin{pmatrix}7\\-5\\3\end{pmatrix}}\wedge {\begin{pmatrix}1\\3\\-4\end{pmatrix}}$

in ${}\bigwedge ^{2}\mathbb {R} ^{3}$ as a linear combination of the wedge products ${}e_{1}\wedge e_{2}$, ${}e_{1}\wedge e_{3}$, and ${}e_{2}\wedge e_{3}$.

Exercise (2 marks)

Express the wedge product ${}{\begin{pmatrix}-2\\5\\-4\end{pmatrix}}\wedge {\begin{pmatrix}7\\-2\\4\end{pmatrix}}$ in the standard basis of ${}\bigwedge ^{2}\mathbb {R} ^{3}$.

Exercise (4 marks)

Express the wedge product

$-2{\begin{pmatrix}3\\6\\-2\\5\end{pmatrix}}\wedge {\begin{pmatrix}2\\7\\4\\0\end{pmatrix}}\wedge {\begin{pmatrix}0\\3\\-4\\-2\end{pmatrix}}+4{\begin{pmatrix}1\\2\\3\\4\end{pmatrix}}\wedge {\begin{pmatrix}1\\-1\\-2\\3\end{pmatrix}}\wedge {\begin{pmatrix}7\\6\\5\\-4\end{pmatrix}}$

in the standard basis of ${}\bigwedge ^{3}\mathbb {R} ^{4}$.

Exercise (5 marks)

We consider the basis

$v_{1}={\begin{pmatrix}9\\8\\1\end{pmatrix}},\,v_{2}={\begin{pmatrix}4\\7\\-3\end{pmatrix}},\,v_{3}={\begin{pmatrix}2\\5\\-2\end{pmatrix}}$

of ${}\mathbb {R} ^{3}$, and the corresponding induced basis

${}{\mathfrak {v}}=v_{1}\wedge v_{2},\,v_{1}\wedge v_{3},\,v_{2}\wedge v_{3}\,$

of ${}\bigwedge ^{2}\mathbb {R} ^{3}$. Determine the base change matrices (in both directions) betwee the basis ${}{\mathfrak {v}}$ and the standard basis ${}e_{1}\wedge e_{2},\,e_{1}\wedge e_{3},\,e_{2}\wedge e_{3}$.

Exercise (5 marks)

Let

$\varphi \colon \mathbb {R} ^{3}\longrightarrow \mathbb {R} ^{3}$

be the linear mapping given by the matrix

${\begin{pmatrix}4&-2&5\\6&8&-3\\1&4&-1\end{pmatrix}}.$

Determine the matrix of ${}\bigwedge ^{2}\varphi$ with respect to the standard bases of the wedge products.

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