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 $V^{\ast }$, the evaluation mapping

$V\times V^{\ast }⟶K,(v,f)⟼f(v),$

is bilinear.

Exercises

Exercise

Compute the determinant of the matrix

$\left(\begin{array}{cc}1+3\mathrm{i}& 5-\mathrm{i}\\ 3-2\mathrm{i}& 4+\mathrm{i}\end{array}\right).$

Exercise

Compute the determinant of the matrix

$\left(\begin{array}{ccc}1& 3& 5\\ 2& 1& 3\\ 8& 7& 4\end{array}\right).$

Exercise

We consider the matrix

$\left(\begin{array}{ccc}2& -1& -1\\ -1& 2& -1\\ -1& -1& 2\end{array}\right).$

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

$\phi :V⟶W$

denote a $K$-linear mapping. Show that $\phi$ is multilinear and alternating.

Exercise

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

$K\times K=K^2⟶K,(a,b)⟼a\cdot b,$

is multilinear. Is it also alternating?

Exercise

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

$K^n\times K^n⟶K,(\left(\begin{array}{c}{u}_1\\ ⋮\\ u_n\end{array}\right),\left(\begin{array}{c}{v}_1\\ ⋮\\ v_n\end{array}\right))⟼(u_1,\dots ,u_n)\circ \left(\begin{array}{c}{v}_1\\ ⋮\\ v_n\end{array}\right),$

is multilinear.

Exercise

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

$\mathrm{Map}(I,K)\times \mathrm{Map}(J,K)⟶\mathrm{Map}(I\times J,K),(f,g)⟼f\otimes g,$

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

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

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

$\mathrm{trace}\left(M\right)=1+detM-det(E_2-M)$

holds.

Exercise

Let $z\in ℂ$ and let

$ℂ⟶ℂ,w⟼zw,$

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

${ℝ}^2\to {ℝ}^2$.

Exercise

Let $K$ be a field, and let $V_1,\dots ,V_n$ and $W$ be vector spaces over $K$. Let

$\Phi :V_1\times \cdots \times V_n⟶W$

be a multilinear mapping, and let $v_{i1},\dots ,v_{i{m}_i}\in V_i$ and $a_{ij}\in K$ for $i=1,\dots ,n$ and $j=1,\dots ,m_i$. Show that

$\Phi (\sum _{j=1}^{m_1}{a}_{1j}{v}_{1j},\dots ,\sum _{j=1}^{m_n}{a}_{nj}{v}_{nj})=\sum _{(j_1,\dots ,j_n)\in \{1,\dots ,m_1\}\times \cdots \times \{1,\dots ,m_n\}}{a}_{1j_1}\cdots a_{n{j}_n}\Phi (v_{1j_1},\dots ,v_{n{j}_n})$

holds.

Exercise

Let $K$ be a field, and let $V_1,\dots ,V_n$ and $W$ denote vector spaces over $K$. Let $v_{i_j}$, $i_j\in I_j$, be generating systems of $V_j$, $j=1,\dots ,n$. Show that a multilinear mapping

$\mathrm{△}:V_1\times \cdots \times V_n⟶W$

is determined by

$\mathrm{△}(v_{i_1},\dots ,v_{i_n}).$

Exercise

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

$\mathrm{△}:V\times V⟶K$

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

$\mathrm{△}\left(\begin{array}{c}u+2v\\ v+3w\end{array}\right).$

Exercise

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

${\mathrm{Mat}}_2(K)⟶K,\left(\begin{array}{cc}a& b\\ c& d\end{array}\right)⟼ad+cb,$

is multilinear, but not alternating.

Exercise

Let $K$ be a field. Is the mapping

${\mathrm{Mat}}_2(K)⟶K,\left(\begin{array}{cc}a& b\\ c& d\end{array}\right)⟼ac-bd,$

multilinear in den rows? In the columns?

Exercise

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

${\mathrm{Mat}}_n(K)=(K^n{)}^n⟶K,M⟼detM,$

fulfills (for arbitrary $k\in \{1,\dots ,n\}$ and arbitrary $n-1$ vectors $v_1,\dots ,v_{k-1},v_{k+1},\dots ,v_n\in K^n$, for $u\in K^n$ and for $s\in K$) the equality

$det\left(\begin{array}{c}{v}_1\\ ⋮\\ v_{k-1}\\ su\\ v_{k+1}\\ ⋮\\ v_n\end{array}\right)=sdet\left(\begin{array}{c}{v}_1\\ ⋮\\ v_{k-1}\\ u\\ v_{k+1}\\ ⋮\\ v_n\end{array}\right).$

Exercise

Let $M$ be the following square matrix

$M=\left(\begin{array}{cc}A& B\\ 0& D\end{array}\right),$

where $A$ and $D$ are square matrices. Prove that $detM=detA\cdot detD$.

Exercise

Let $M$ be a square matrix of the form

$M=\left(\begin{array}{cc}A& B\\ C& D\end{array}\right)$

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

$detM=detA\cdot detD-detB\cdot detC$

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

${\mathrm{Hom}}_K(V,W)\times V⟶W,(\phi ,v)⟼\phi (v),$

is multilinear.

Exercise

Let $K$ be a field, and let $V_1,\dots ,V_n$ and $W$ denote vector spaces over $K$. Let

$\Phi :V_1\times \cdots \times V_n⟶W$

be a multilinear mapping. Show that the set

$\{(v_1,\dots ,v_n)\in V_1\times \cdots \times V_n\mid \Phi (v_1,\dots ,v_n)=0\}$

is, in general, not a linear subspace of $V_1\times \cdots \times V_n$.

Exercise

Let $K$ be a field, and let $V_1,\dots ,V_n$ 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 ${\mathrm{Mult}}_K(V_1,\dots ,V_n;W)$.

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 ${\mathrm{Alt}}_K^n(V,W)$) is a linear subspace of ${\mathrm{Mult}}_K(V,\dots ,V;W)$ (where the vector space $V$ appears $n$-fold).

Exercise

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

$\phi :V⟶W$

denote a $K$-linear mapping, Let

$\mathrm{△}:W^m⟶K$

denote a multilinear mapping. Show that the composed mapping

$V^m⟶K,(v_1,\dots ,v_m)⟼\mathrm{△}(\phi (v_1),\dots ,\phi (v_m)),$

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

Exercise

Let $K$ be a field, and let $V_1,\dots ,V_n,W_1,\dots ,W_n$ and $Z$ be vector spaces over $K$. Let

${\phi }_i:V_i⟶W_i$

denote linear mappings, and let

$\pi :W_1\times \cdots \times W_n⟶Z$

be a multilinear mapping. Show that the mapping

$\pi \circ ({\phi }_1\times \cdots \times {\phi }_n):V_1\times \cdots \times V_n⟶Z,(v_1,\dots ,v_n)⟼\pi ({\phi }_1(v_1),\dots ,{\phi }_n(v_n)),$

is also multilinear.

Exercise

Compute for the (complex) matrix

$M=\left(\begin{array}{ccc}1+\mathrm{i}& 2\mathrm{i}& 3\\ 0& 1-\mathrm{i}& -1+3\mathrm{i}\\ 4-\mathrm{i}& 0& 2\end{array}\right)$

the determinant and the inverse matrix.

Exercise *

Determine for which $x\in ℂ$ the matrix

$\left(\begin{array}{cc}{x}^2+x& -x\\ -x^3+2x^2+5x-1& x^2-x\end{array}\right)$

is invertible.

The Christmas exercise for the whole family

Exercise

Which construction principle is behind the sequence

$1,11,21,1211,111221,312211,\mathrm{...}?$

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

Exercise (2 marks)

Compute the determinant of the elementary matrices.

Exercise (3 marks)

Compute the determinant of the matrix

$\left(\begin{array}{ccc}1+\mathrm{i}& 3-2\mathrm{i}& 5\\ \mathrm{i}& 1& 3-\mathrm{i}\\ 2\mathrm{i}& -4-\mathrm{i}& 2+\mathrm{i}\end{array}\right).$

Exercise (3 marks)

Compute the determinant of the matrix

$A=\left(\begin{array}{cccc}2& 1& 0& -2\\ 1& 3& 3& -1\\ 3& 2& 4& -3\\ 2& -2& 2& 3\end{array}\right).$

Exercise (3 marks)

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

$\mathrm{△}:V\times V\times V⟶K$

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

$\mathrm{△}\left(\begin{array}{c}u+v+w\\ 2u+3z\\ 4w-5z\end{array}\right).$

Exercise (3 marks)

Let $K$ be a field, and let $V_1,\dots ,V_n$ vector spaces over $K$. Let

${\phi }_i:V_i⟶K$

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

$\phi :V_1\times \cdots \times V_n⟶K,(v_1,\dots ,v_n)⟼{\phi }_1(v_1)\cdots {\phi }_n(v_n),$

is multilinear.

Exercise (3 (1+2) marks)

Let $K$ be a field, and let

$T=\{M\in {\mathrm{Mat}}_{3\times 3}(K)\mid detM=0\}\subset {\mathrm{Mat}}_{3\times 3}(K)=K^9$

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

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

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

Exercise (4 marks)

We consider the mapping

$f:\mathbb{N}⟶\mathbb{N}$

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

$T=\{M\in {\mathrm{Mat}}_{3\times 3}(K)\mid detM=0\}\subset {\mathrm{Mat}}_{3\times 3}(K)=K^9$

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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