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

Exercises

Exercise

Compute the following product of matrices

${\begin{pmatrix}Z&E&I&L&E\\R&E&I&H&E\\H&O&R&I&Z\\O&N&T&A&L\end{pmatrix}}\cdot {\begin{pmatrix}S&E&I\\P&V&K\\A&E&A\\L&R&A\\T&T&L\end{pmatrix}}.$

Exercise *

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

${\begin{pmatrix}2-{\mathrm {i} }&-1-3{\mathrm {i} }&-1\\{\mathrm {i} }&0&4-2{\mathrm {i} }\end{pmatrix}}{\begin{pmatrix}1+{\mathrm {i} }\\1-{\mathrm {i} }\\2+5{\mathrm {i} }\end{pmatrix}}.$

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 ${}Me_{j}$ 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

${}D={\begin{pmatrix}d_{11}&0&\cdots &\cdots &0\\0&d_{22}&0&\cdots &0\\\vdots &\ddots &\ddots &\ddots &\vdots \\0&\cdots &0&d_{n-1\,n-1}&0\\0&\cdots &\cdots &0&d_{nn}\end{pmatrix}}\,$

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

Exercise

Let

${}D={\begin{pmatrix}d_{11}&0&\cdots &\cdots &0\\0&d_{22}&0&\cdots &0\\\vdots &\ddots &\ddots &\ddots &\vdots \\0&\cdots &0&d_{n-1\,n-1}&0\\0&\cdots &\cdots &0&d_{nn}\end{pmatrix}}\,$

be a diagonal matrix, and let ${}c={\begin{pmatrix}c_{1}\\\vdots \\c_{n}\end{pmatrix}}$ be an ${}n$-tuple over a field ${}K$, and let ${}x={\begin{pmatrix}x_{1}\\\vdots \\x_{n}\end{pmatrix}}$ 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

${\begin{pmatrix}2+{\mathrm {i} }&1-{\frac {1}{2}}{\mathrm {i} }&4{\mathrm {i} }\\-5+7{\mathrm {i} }&{\sqrt {2}}+{\mathrm {i} }&0\end{pmatrix}}{\begin{pmatrix}-5+4{\mathrm {i} }&3-2{\mathrm {i} }\\{\sqrt {2}}-{\mathrm {i} }&e+\pi {\mathrm {i} }\\1&-{\mathrm {i} }\end{pmatrix}}{\begin{pmatrix}1+{\mathrm {i} }\\2-3{\mathrm {i} }\end{pmatrix}},$

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

${}M={\begin{pmatrix}2&4&6\\1&3&5\\0&1&2\end{pmatrix}}\,,$

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

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

Exercise

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

$(cx,cy),\,c\in \mathbb {R} .$

Exercise

Draw two points ${}P$ and ${}Q$ in the Cartesian plane ${}\mathbb {R} ^{2}$ 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 ${}s_{1},\ldots ,s_{k}\in K$ and ${}v_{1},\ldots ,v_{n}\in V$. Show

${}{\left(\sum _{i=1}^{k}s_{i}\right)}\cdot {\left(\sum _{j=1}^{n}v_{j}\right)}=\sum _{1\leq i\leq k,\,1\leq j\leq n}s_{i}\cdot v_{j}\,.$

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 ${}\mathbb {R} ^{2}$ are linear subspaces:

1. ${}V_{1}={\left\{(x,y)\in \mathbb {R} ^{2}\mid x+2y=0\right\}}$,
2. ${}V_{2}={\left\{(x,y)\in \mathbb {R} ^{2}\mid x\geq y\right\}}$,
3. ${}V_{3}={\left\{(x,y)\in \mathbb {R} ^{2}\mid y=x+1\right\}}$,
4. ${}V_{4}={\left\{(x,y)\in \mathbb {R} ^{2}\mid xy=0\right\}}$.

Exercise

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

${\begin{matrix}a_{11}x_{1}+a_{12}x_{2}+\cdots +a_{1n}x_{n}&=&0\\a_{21}x_{1}+a_{22}x_{2}+\cdots +a_{2n}x_{n}&=&0\\\vdots &\vdots &\vdots \\a_{m1}x_{1}+a_{m2}x_{2}+\cdots +a_{mn}x_{n}&=&0\end{matrix}}$

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

${\begin{pmatrix}a_{11}&a_{12}\\a_{21}&a_{22}\end{pmatrix}},$

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

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

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

Exercise

Let

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

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

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

Exercise

Show that the subset

${}S={\left\{f:\mathbb {R} \rightarrow \mathbb {R} \mid f{\text{ continuous}}\right\}}\subseteq \operatorname {Map} \,{\left(\mathbb {R} ,\mathbb {R} \right)}\,$

is a linear subspace.

Exercise

Show that the subset

${}S={\left\{f:\mathbb {R} \rightarrow \mathbb {R} \mid f{\text{ differentiable}}\right\}}\subseteq \operatorname {Map} \,{\left(\mathbb {R} ,\mathbb {R} \right)}\,$

is a linear subspace.

Exercise

Show that the subset

${}M={\left\{f:\mathbb {R} \rightarrow \mathbb {R} \mid f{\text{ monotonic}}\right\}}\subseteq \operatorname {Map} \,{\left(\mathbb {R} ,\mathbb {R} \right)}\,$

is not a linear subspace.

Hand-in-exercises

Exercise (3 marks)

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

${\begin{pmatrix}3-2{\mathrm {i} }&1+5{\mathrm {i} }&0\\7{\mathrm {i} }&2+{\mathrm {i} }&4-{\mathrm {i} }\end{pmatrix}}{\begin{pmatrix}1-2{\mathrm {i} }&-{\mathrm {i} }\\3-4{\mathrm {i} }&2+3{\mathrm {i} }\\5-7{\mathrm {i} }&2-{\mathrm {i} }\end{pmatrix}}.$

Exercise (3 marks)

We consider the matrix

${}M={\begin{pmatrix}0&a&b&c\\0&0&d&e\\0&0&0&f\\0&0&0&0\end{pmatrix}}\,$

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

${\begin{pmatrix}a&b\\0&c\end{pmatrix}}.$

Exercise (2 marks)

Find, appart from the matrices ${}{\begin{pmatrix}1&0\\0&1\end{pmatrix}}$ and ${}{\begin{pmatrix}-1&0\\0&-1\end{pmatrix}}$, four more matrices ${}M$ fulfilling the property ${}M^{2}={\begin{pmatrix}1&0\\0&1\end{pmatrix}}$.

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 ${}s\neq 0$ and ${}v\neq 0$, then ${}sv\neq 0$.

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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