Mathematics for Applied Sciences (Osnabrück 2023-2024)/Part I/Exercise sheet 22

Exercises

Exercise

Compute the following product of matrices

$\left(\begin{array}{ccccc}Z& E& I& L& E\\ R& E& I& H& E\\ H& O& R& I& Z\\ O& N& T& A& L\end{array}\right)\cdot \left(\begin{array}{ccc}S& E& I\\ P& V& K\\ A& E& A\\ L& R& A\\ T& T& L\end{array}\right).$

Exercise *

Compute, over the complex numbers, the following product of matrices

$\left(\begin{array}{ccc}2-\mathrm{i}& -1-3\mathrm{i}& -1\\ \mathrm{i}& 0& 4-2\mathrm{i}\end{array}\right)\left(\begin{array}{c}1+\mathrm{i}\\ 1-\mathrm{i}\\ 2+5\mathrm{i}\end{array}\right).$

Exercise

Determine the product of matrices

$e_i\circ e_j,$

where the $i$-th standard vector (of length $n$) is considered as a row vector, and the $j$-th standard vector (also of length $n$) is considered as a column vector.

Exercise

Let $M$ be an $m\times n$- matrix. Show that the matrix product $M{e}_j$ of $M$ with the $j$-th standard vector (regarded as a column vector) is the $j$-th column of $M$. What is $e_{i}M$, where $e_i$ is the $i$-th standard vector (regarded as a row vector)?

Exercise

Let

$D=\left(\begin{array}{ccccc}{d}_{11}& 0& \cdots & \cdots & 0\\ 0& d_{22}& 0& \cdots & 0\\ ⋮& \ddots & \ddots & \ddots & ⋮\\ 0& \cdots & 0& d_{n-1n-1}& 0\\ 0& \cdots & \cdots & 0& d_{nn}\end{array}\right)$

be a diagonal matrix, and $M$ an $n\times n$-matrix. Describe $DM$ and $MD$.

Exercise

Let

$D=\left(\begin{array}{ccccc}{d}_{11}& 0& \cdots & \cdots & 0\\ 0& d_{22}& 0& \cdots & 0\\ ⋮& \ddots & \ddots & \ddots & ⋮\\ 0& \cdots & 0& d_{n-1n-1}& 0\\ 0& \cdots & \cdots & 0& d_{nn}\end{array}\right)$

be a diagonal matrix, and let $c=\left(\begin{array}{c}{c}_1\\ ⋮\\ c_n\end{array}\right)$ be an $n$-tuple over a field $K$, and let $x=\left(\begin{array}{c}{x}_1\\ ⋮\\ x_n\end{array}\right)$ be a tuple of variables. What is specific about the system of linear equations

$Dx=c,$

and how can you solve it?

Exercise

Compute the product of matrices

$\left(\begin{array}{ccc}2+\mathrm{i}& 1-\frac{1}{2}\mathrm{i}& 4\mathrm{i}\\ -5+7\mathrm{i}& \sqrt{2}+\mathrm{i}& 0\end{array}\right)\left(\begin{array}{cc}-5+4\mathrm{i}& 3-2\mathrm{i}\\ \sqrt{2}-\mathrm{i}& e+\pi \mathrm{i}\\ 1& -\mathrm{i}\end{array}\right)\left(\begin{array}{c}1+\mathrm{i}\\ 2-3\mathrm{i}\end{array}\right),$

according to the two possible parentheses.

For the following statement we will get soon a simpler proof via the relation between matrices and linear mappings.

Exercise

Show that the multiplication of matrices is associative. More precisely: Let $K$ be a field, and let $A$ be an $m\times n$-matrix, $B$ an $n\times p$-matrix, and $C$ a $p\times r$-matrix over $K$. Show that $(AB)C=A(BC)$.

Exercise

Show that the matrix multiplication of square matrices is, in general, not commutative.

For a matrix $M$, we denote by $M^n$ the $n$-th fold composition (matrix multiplication) of $M$ with itself. $M^n$ is called the $n$-th power of the matrix.

Exercise

Compute, for the matrix

$M=\left(\begin{array}{ccc}2& 4& 6\\ 1& 3& 5\\ 0& 1& 2\end{array}\right),$

the powers

$M^i,i=1,2,3,4.$

Exercise

Out of the resources $R_1,R_2$, and $R_3$, several commodities $P_1,P_2,P_3,P_4$ are produced. The following table shows how much of the resources are needed to produce the commodities (always in suitable units).

	$R_1$	$R_2$	$R_3$
$P_1$	$6$	$2$	$3$
$P_2$	$4$	$1$	$2$
$P_3$	$0$	$5$	$2$
$P_4$	$2$	$1$	$5$

a) Establish a matrix that computes, applied to a four-tuple of commodities, the required resources.

b) The following table shows how much of each commodity shall be produced in a month.

	$P_1$	$P_2$	$P_3$	$P_4$
	$6$	$4$	$7$	$5$

What resources are necessary?

c) The following table shows how much of each resource is delivered on a certain day.

	$R_1$	$R_2$	$R_3$
	$12$	$9$	$13$

What tuples of commodities can be produced from this without waste?

Exercise

Determine (approximately) the coordinates of the sketched point (the side length of a box represents a unit).

Exercise

Draw the following points in the Cartesian plane ${ℝ}^2$.

$(3,-7),(-1,-2),(0,5),(4,4),(4,5),(-3,0),(0,0).$

Exercise

Let a point $P=(x,y)$ be given in the plane ${ℝ}^2$. Sketch the points

$(-x,y),(x,-y),(-x,-y),(3x,3y),(-2x,-2y).$

Exercise

Let a point $P=(x,y)$ be given in the plane ${ℝ}^2$. Sketch the set of all points

$(cx,cy),c\in ℝ.$

Exercise

Draw two points $P$ and $Q$ in the Cartesian plane ${ℝ}^2$ and add them.

Exercise

Show that the product space $K^n$, for a field $K$, is, with componentwise addition and scalar multiplication, the properties

1. $r(su)=(rs)u$,
2. $r(u+v)=ru+rv$,
3. $(r+s)u=ru+su$,

4. $1u=u,$

hold.

Exercise

Let $K$ be a field, and let $V$ and $W$ be vector spaces over $K$. Show that the product set

$V\times W$

is also a $K$-vector space.

Exercise

Let $V$ be a vector space over a field $K$. Let $s_1,\dots ,s_k\in K$ and $v_1,\dots ,v_n\in V$. Show

$\left(\sum _{i=1}^k{s}_i\right)\cdot \left(\sum _{j=1}^n{v}_j\right)=\sum _{1\le i\le k,1\le j\le n}{s}_i\cdot v_j.$

Exercise

Show that the addition and the scalar multiplication of a vector space $V$ can be restricted to a linear subspace, and that this subspace with the inherited structures of $V$ is a vector space itself.

Exercise

Check whether the following subsets of ${ℝ}^2$ are linear subspaces:

1. $V_1=\{(x,y)\in {ℝ}^2\mid x+2y=0\}$,
2. $V_2=\{(x,y)\in {ℝ}^2\mid x\ge y\}$,
3. $V_3=\{(x,y)\in {ℝ}^2\mid y=x+1\}$,
4. $V_4=\{(x,y)\in {ℝ}^2\mid xy=0\}$.

Exercise

Let $K$ be a field, and let

$\begin{array}{ccc}{a}_{11}{x}_1+a_{12}{x}_2+\cdots +a_{1n}{x}_n& =& 0\\ a_{21}{x}_1+a_{22}{x}_2+\cdots +a_{2n}{x}_n& =& 0\\ ⋮& ⋮& ⋮\\ a_{m1}{x}_1+a_{m2}{x}_2+\cdots +a_{mn}{x}_n& =& 0\end{array}$

be a system of linear equations over $K$. Show that the set of all solutions of this system is a linear subspace of $K^n$. How is this solution space related to the solution spaces of the individual equations?

Exercise

Let $D$ be the set of all real $2\times 2$-matrices

$\left(\begin{array}{cc}{a}_{11}& a_{12}\\ a_{21}& a_{22}\end{array}\right),$

which fulfill the condition

$a_{11}{a}_{22}-a_{21}{a}_{12}=0.$

Show that $D$ is not a linear subspace in the space of all $2\times 2$-matrices.

Exercise

Let $K$ be a field, and let $V$ be a $K$-vector space. Let $U,W\subseteq V$ be linear subspaces of $V$. Prove that the union $U\cup W$ is a linear subspace of $V$ if and only if $U\subseteq W$ or $W\subseteq U$.

Exercise

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

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

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

Exercise

Let

$C=\{{\left(x_n\right)}_{n\in \mathbb{N}}\mid \text{Cauchy sequence in }ℝ\}$

be the set of all real Cauchy sequences. Show that $C$ is a linear subspace of the space of all sequences

$F=\{{\left(x_n\right)}_{n\in \mathbb{N}}\mid \text{sequence in }ℝ\}.$

Exercise

Show that the subset

$S=\{f:ℝ\to ℝ\mid f\text{ continuous}\}\subseteq \mathrm{Map}(ℝ,ℝ)$

is a linear subspace.

Exercise

Show that the subset

$S=\{f:ℝ\to ℝ\mid f\text{ differentiable}\}\subseteq \mathrm{Map}(ℝ,ℝ)$

is a linear subspace.

Exercise

Show that the subset

$M=\{f:ℝ\to ℝ\mid f\text{ monotonic}\}\subseteq \mathrm{Map}(ℝ,ℝ)$

is not a linear subspace.

Hand-in-exercises

Exercise (3 marks)

Compute, over the complex numbers, the following product of matrices

$\left(\begin{array}{ccc}3-2\mathrm{i}& 1+5\mathrm{i}& 0\\ 7\mathrm{i}& 2+\mathrm{i}& 4-\mathrm{i}\end{array}\right)\left(\begin{array}{cc}1-2\mathrm{i}& -\mathrm{i}\\ 3-4\mathrm{i}& 2+3\mathrm{i}\\ 5-7\mathrm{i}& 2-\mathrm{i}\end{array}\right).$

Exercise (3 marks)

We consider the matrix

$M=\left(\begin{array}{cccc}0& a& b& c\\ 0& 0& d& e\\ 0& 0& 0& f\\ 0& 0& 0& 0\end{array}\right)$

over a field $K$. Show that the fourth power of $M$ is $0$, that is,

$M^4=MMMM=0.$

Exercise (4 marks)

Let $n\in \mathbb{N}$. Find and prove a formula for the $n$-th power of the matrix

$\left(\begin{array}{cc}a& b\\ 0& c\end{array}\right).$

Exercise (2 marks)

Find, appart from the matrices $\left(\begin{array}{cc}1& 0\\ 0& 1\end{array}\right)$ and $\left(\begin{array}{cc}-1& 0\\ 0& -1\end{array}\right)$, four more matrices $M$ fulfilling the property $M^2=\left(\begin{array}{cc}1& 0\\ 0& 1\end{array}\right)$.

Exercise (3 marks)

Let $K$ be a field, and let $V$ be a $K$-vector space. Show that the following properties hold (for $v\in V$ and $s\in K$).

1. We have $0v=0$.
2. We have $s0=0$.
3. We have $(-1)v=-v$.
4. If $s\ne 0$ and $v\ne 0$, then $sv\ne 0$.

Exercise (3 marks)

Give an example of a vector space $V$ and of three subsets of $V$ that satisfy two of the subspace axioms, but not the third.

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