Linear algebra (Osnabrück 2024-2025)/Part I/Exercise sheet 16

Exercise for the break

Exercise

Show that, for a ${}K$-vector space ${}V$ with dual space ${}{V}^{*}$, the evaluation mapping

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

is bilinear.

Exercises

Exercise

Compute the determinant of the matrix

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

Exercise

Compute the determinant of the matrix

${\begin{pmatrix}1&3&5\\2&1&3\\8&7&4\end{pmatrix}}.$

Exercise

We consider the matrix

${\begin{pmatrix}2&-1&-1\\-1&2&-1\\-1&-1&2\end{pmatrix}}.$

1. Compute the determinant of ${}M$.
2. Determine the determinant for every matrix that arises when we remove from ${}M$ a row and a column.

Exercise

Prove by induction that the determinant of an upper triangular matrix is equal to the product of the diagonal elements.

Exercise

Prove by induction that the determinant of a lower triangular matrix is equal to the product of the diagonal elements.

Exercise

Let ${}K$ be a field, let ${}V$ and ${}W$ be ${}K$-vector spaces, and let

$\varphi \colon V\longrightarrow W$

denote a ${}K$-linear mapping. Show that ${}\varphi$ is multilinear and alternating.

Exercise

Let ${}K$ be a field. Show that the multiplication

$K\times K=K^{2}\longrightarrow K,(a,b)\longmapsto a\cdot b,$

is multilinear. Is it also alternating?

Exercise

Let ${}K$ be a field, and let ${}n\in \mathbb {N}$. Show that the mapping

$K^{n}\times K^{n}\longrightarrow K,({\begin{pmatrix}u_{1}\\\vdots \\u_{n}\end{pmatrix}},{\begin{pmatrix}v_{1}\\\vdots \\v_{n}\end{pmatrix}})\longmapsto (u_{1},\ldots ,u_{n})\circ {\begin{pmatrix}v_{1}\\\vdots \\v_{n}\end{pmatrix}},$

is multilinear.

Exercise

Let ${}K$ be a field, and let ${}I$ and ${}J$ denote finite index sets. Show that the mapping

$\operatorname {Map} \,{\left(I,K\right)}\times \operatorname {Map} \,{\left(J,K\right)}\longrightarrow \operatorname {Map} \,{\left(I\times J,K\right)},(f,g)\longmapsto f\otimes g,$

given by

${}(f\otimes g)(i,j):=f(i)\cdot g(j)\,,$

is multilinear.

Exercise

Check the multilinearity and the property to be alternating, directly for the determinant of a ${}2\times 2$-matrix.

Exercise

Check the multilinearity and the property to be alternating, directly for the determinant of a ${}3\times 3$-matrix.

Exercise

Show that, for every elementary matrix ${}E$, the relation

${}\det E=\det {E^{\text{tr}}}\,$

holds.

Exercise

Use the image to convince yourself that, given two vectors ${}(x_{1},y_{1})$ and ${}(x_{2},y_{2})$, the determinant of the ${}2\times 2$-matrix defined by these vectors is equal (up to sign) to the area of the plane parallelogram spanned by the vectors.

Exercise

Let ${}M$ be a ${}2\times 2$-matrix. Show that

${}\operatorname {trace} {\left(M\right)}=1+\det M-\det {\left(E_{2}-M\right)}\,$

holds.

Exercise

Let ${}z\in \mathbb {C}$ and let

$\mathbb {C} \longrightarrow \mathbb {C} ,w\longmapsto zw,$

be the associated multiplication. Compute the determinant of this map, considering it as a real-linear map

${}\mathbb {R} ^{2}\rightarrow \mathbb {R} ^{2}$.

Exercise

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

$\Phi \colon V_{1}\times \cdots \times V_{n}\longrightarrow W$

be a multilinear mapping, and let ${}v_{1j},\ldots ,v_{m_{j}j}\in V_{j}$ and ${}a_{ij}\in K$. Show that

$\Phi {\left(\sum _{i=1}^{m_{1}}a_{i1}v_{i1},\ldots ,\sum _{i=1}^{m_{n}}a_{in}v_{in}\right)}=\sum _{(i_{1},\ldots ,i_{n})\in \{1,\ldots ,m_{1}\}\times \cdots \times \{1,\ldots ,m_{n}\}}a_{i_{1}1}\cdots a_{i_{n}n}\Phi {\left(v_{i_{1}1},\ldots ,v_{i_{n}n}\right)}\,$

holds.

Exercise

Let ${}K$ be a field, and let ${}V_{1},\ldots ,V_{n}$ and ${}W$ denote vector spaces over ${}K$. Let ${}v_{i_{j}}$, ${}i_{j}\in I_{j}$, be generating systems of ${}V_{j}$, ${}j=1,\ldots ,n$. Show that a multilinear mapping

$\triangle \colon V_{1}\times \cdots \times V_{n}\longrightarrow W$

is determined by

$\triangle (v_{i_{1}},\ldots ,v_{i_{n}}).$

Exercise

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

$\triangle \colon V\times V\longrightarrow K$

be a multilinear and alternating mapping. Let ${}u,v,w\in V$. Simplify

$\triangle {\begin{pmatrix}u+2v\\v+3w\end{pmatrix}}.$

Exercise

Let ${}K$ be a field. Show that the mapping

$\operatorname {Mat} _{2}(K)\longrightarrow K,{\begin{pmatrix}a&b\\c&d\end{pmatrix}}\longmapsto ad+cb,$

is multilinear, but not alternating.

Exercise

Let ${}K$ be a field. Is the mapping

$\operatorname {Mat} _{2}(K)\longrightarrow K,{\begin{pmatrix}a&b\\c&d\end{pmatrix}}\longmapsto ac-bd,$

multilinear in den rows? In the columns?

Exercise

Let ${}K$ be a field and ${}n\in \mathbb {N} _{+}$. Show that the determinant

$\operatorname {Mat} _{n}(K)=(K^{n})^{n}\longrightarrow K,M\longmapsto \det M,$

fulfills (for arbitrary ${}k\in \{1,\ldots ,n\}$ and arbitrary ${}n-1$ vectors ${}v_{1},\ldots ,v_{k-1},v_{k+1},\ldots ,v_{n}\in K^{n}$, for ${}u\in K^{n}$ and for ${}s\in K$) the equality

${}\det {\begin{pmatrix}v_{1}\\\vdots \\v_{k-1}\\su\\v_{k+1}\\\vdots \\v_{n}\end{pmatrix}}=s\det {\begin{pmatrix}v_{1}\\\vdots \\v_{k-1}\\u\\v_{k+1}\\\vdots \\v_{n}\end{pmatrix}}\,.$

Exercise

Let ${}M$ be the following square matrix

${}M={\begin{pmatrix}A&B\\0&D\end{pmatrix}}\,,$

where ${}A$ and ${}D$ are square matrices. Prove that ${}\det M=\det A\cdot \det D$.

Exercise

Let ${}M$ be a square matrix of the form

${}M={\begin{pmatrix}A&B\\C&D\end{pmatrix}}\,$

with square matrices ${}A,B,C$ and ${}D$. Show by an example that the equality

${}\det M=\det A\cdot \det D-\det B\cdot \det C\,$

does not hold in general.

Exercise

Let ${}K$ be a field, and let ${}V$ and ${}W$ denote a ${}K$-vector space. Determine whether the mapping

$\operatorname {Hom} _{K}{\left(V,W\right)}\times V\longrightarrow W,(\varphi ,v)\longmapsto \varphi (v),$

is multilinear.

Exercise

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

$\Phi \colon V_{1}\times \cdots \times V_{n}\longrightarrow W$

be a multilinear mapping. Show that the set

${\left\{\left(v_{1},\,\ldots ,\,v_{n}\right)\in V_{1}\times \cdots \times V_{n}\mid \Phi \left(v_{1},\,\ldots ,\,v_{n}\right)=0\right\}}$

is, in general, not a linear subspace of ${}V_{1}\times \cdots \times V_{n}$.

Exercise

Let ${}K$ be a field, and let ${}V_{1},\ldots ,V_{n}$ and ${}W$ denote vector spaces over ${}K$. Show that the set of all multilinear mappings is, in a natural way, a vector space, denoted by ${}\operatorname {Mult} _{K}{\left(V_{1},\ldots ,V_{n},W\right)}$.

Exercise

Let ${}K$ be a field, let ${}V$ and ${}W$ be vector spaces over ${}K$, and ${}n\in \mathbb {N}$. Show that the set of all alternating mappings (denoted by ${}\operatorname {Alt} _{K}^{n}{\left(V,W\right)}$) is a linear subspace of ${}\operatorname {Mult} _{K}{\left(V,\ldots ,V,W\right)}$ (where the vector space ${}V$ appears ${}n$-fold).

Exercise

Let ${}K$ be a field, let ${}V$ and ${}W$ be ${}K$-vector spaces, and let

$\varphi \colon V\longrightarrow W$

denote a ${}K$-linear mapping, Let

$\triangle \colon W^{m}\longrightarrow K$

denote a multilinear mapping. Show that the composed mapping

$V^{m}\longrightarrow K,(v_{1},\ldots ,v_{m})\longmapsto \triangle \left(\varphi (v_{1}),\,\ldots ,\,\varphi (v_{m})\right),$

is multilinear. Moreover, show that, if ${}\triangle$ is alternating, then also ${}\triangle \circ \varphi ^{n}$ is alternating, and that, if ${}\varphi$ is bijective, also the converse holds.

Exercise

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

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

denote linear mappings, and let

$\pi \colon W_{1}\times \cdots \times W_{n}\longrightarrow Z$

be a multilinear mapping. Show that the mapping

$\pi \circ (\varphi _{1}\times \cdots \times \varphi _{n})\colon V_{1}\times \cdots \times V_{n}\longrightarrow Z,(v_{1},\ldots ,v_{n})\longmapsto \pi (\varphi _{1}(v_{1}),\ldots ,\varphi _{n}(v_{n})),$

is also multilinear.

Exercise

Compute for the (complex) matrix

${}M={\begin{pmatrix}1+{\mathrm {i} }&2{\mathrm {i} }&3\\0&1-{\mathrm {i} }&-1+3{\mathrm {i} }\\4-{\mathrm {i} }&0&2\end{pmatrix}}\,$

the determinant and the inverse matrix.

Exercise *

Determine for which ${}x\in \mathbb {C}$ the matrix

${\begin{pmatrix}x^{2}+x&-x\\-x^{3}+2x^{2}+5x-1&x^{2}-x\end{pmatrix}}$

is invertible.

The Christmas exercise for the whole family

Exercise

Which construction principle is behind the sequence

$1,\,11,\,21,\,1211,\,111221,\,312211,\,...?$

(Some people claim that this exercise is very easy for primary school children, but quite hard for mathematicians.)

Hand-in-exercises

Exercise (2 marks)

Let ${}M\in \operatorname {Mat} _{n}(\mathbb {Q} )$. Show that it does not make a difference, whether we compute the determinant in ${}\mathbb {Q}$, in ${}\mathbb {R}$, or in ${}\mathbb {C}$.

Exercise (2 marks)

Compute the determinant of the elementary matrices.

Exercise (3 marks)

Compute the determinant of the matrix

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

Exercise (3 marks)

Compute the determinant of the matrix

${}A={\begin{pmatrix}2&1&0&-2\\1&3&3&-1\\3&2&4&-3\\2&-2&2&3\end{pmatrix}}\,.$

Exercise (3 marks)

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

$\triangle \colon V\times V\times V\longrightarrow K$

be a multilinear and alternating mapping. Let ${}u,v,w\in V$. Simplify the term

$\triangle {\begin{pmatrix}u+v+w\\2u+3z\\4w-5z\end{pmatrix}}.$

Exercise (3 marks)

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

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

(${}i=1,\ldots ,n$), denote linear mappings. Show that the mapping

$\varphi \colon V_{1}\times \cdots \times V_{n}\longrightarrow K,{\left(v_{1},\ldots ,v_{n}\right)}\longmapsto \varphi _{1}(v_{1})\cdots \varphi _{n}(v_{n}),$

is multilinear.

Exercise (3 (1+2) marks)

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

${}T={\left\{M\in \operatorname {Mat} _{3\times 3}(K)\mid \det M=0\right\}}\subset \operatorname {Mat} _{3\times 3}(K)=K^{9}\,$

be the set of all ${}3\times 3$-matrices with determinant ${}0$.

a) Show that ${}T$ is not a linear subspace of ${}\operatorname {Mat} _{3\times 3}(K)$.

b) Show that ${}T$ contains a linear subspace of ${}\operatorname {Mat} _{3\times 3}(K)$ of dimension ${}6$.

Exercise (4 marks)

We consider the mapping

$f\colon \mathbb {N} \longrightarrow \mathbb {N}$

that is described in Exercise 16.33 (the natural numbers are given as finite sequences in the decimal system).

1. Is ${}f$ increasing?
2. Is ${}f$ surjective?
3. Is ${}f$ injective?
4. Does ${}f$ have a fixed point?

Exercise to give up

Please hand in solutions to the following exercise directly to the lecturer.

Exercise ( marks)

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

${}T={\left\{M\in \operatorname {Mat} _{3\times 3}(K)\mid \det M=0\right\}}\subset \operatorname {Mat} _{3\times 3}(K)=K^{9}\,$

be the set of all ${}3\times 3$-matrices with determinant ${}0$. Does ${}T$ contain a linear subspace of dimension ${}7$?

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