Linear algebra (Osnabrück 2024-2025)/Part I/Exercise sheet 4

Exercise for the break

Exercise

Solve the linear system

$2x+3y=7{\text{ and }}5x+4y=3.$

Exercises

Exercise

Solve the linear equation

${}3x=5\,$

for the following fields ${}K$:

a) ${}K=\mathbb {Q}$,

b) ${}K=\mathbb {R}$,

c) ${}K=\{0,1\}$, the field with two elements from Example 3.8 ,

d) ${}K=\{0,1,2,3,4,5,6\}$, the field with seven elements from Example 3.9 .

The field of complex numbers is introduced in analysis (see also the appendix). A complex number has the form ${}a+b{\mathrm {i} }$ with real numbers ${}a,b$. The multiplication is determined by the rule ${}{\mathrm {i} }\cdot {\mathrm {i} }=-1$. The inverse complex number for ${}a+b{\mathrm {i} }\neq 0$ is ${}{\frac {a}{a^{2}+b^{2}}}-{\frac {b}{a^{2}+b^{2}}}{\mathrm {i} }$.

Exercise

Solve the linear equation

${}(2+5{\mathrm {i} })z=(3-7{\mathrm {i} })\,$

over ${}\mathbb {C}$, and compute the modulus of the solution.

Exercise

Show that the system of linear equations

${\begin{matrix}-4x+6y&=&0\\5x+8y&=&0\,\end{matrix}}$

has only the trivial solution ${}(0,0)$.

Exercise

Does there exist a solution ${}(a,b,c)\in \mathbb {Q} ^{3}$ for the system of linear equations

${}a{\begin{pmatrix}1\\2\\11\\2\end{pmatrix}}+b{\begin{pmatrix}2\\2\\12\\3\end{pmatrix}}+c{\begin{pmatrix}3\\1\\20\\7\end{pmatrix}}={\begin{pmatrix}1\\2\\20\\5\end{pmatrix}}\,$

from Example 4.1 ?

Exercise

Two persons, ${}A$ and ${}B$, are lying beneath a palm tree, ${}A$ has ${}2$ flatbreads, and ${}B$ has ${}3$ flatbreads. A third person ${}C$ joins them, ${}C$ has no flatbread, but ${}5$ thalers. They agree to distribute, against the ${}5$ thalers, the ${}5$ flatbreads uniformly among them. How many thalers gives ${}C$ to ${}A$, how many to ${}B$?

Exercise

The dating service "e-Tarzan meets e-Jane“ is successful, it claims that in each eleven minutes, one of the customers falls in love. How long does it take (in rounded years) until all adult people in Germany (about ${}65000000$) fall in love, if this service is the only possible way.

Exercise

${}M,P,S$ and ${}T$ are the members of one family. In this case, ${}M$ is three times as old as ${}S$ and ${}T$ together, ${}M$ is older than ${}P$, and ${}S$ is older than ${}T$, moreover, the age difference between ${}S$ and ${}T$ is twice as large as the difference between ${}M$ and ${}P$. Furthermore, ${}P$ is seven times as old as ${}T$, and the sum of the ages of all family members is equal to the paternal grandmother's age, which is ${}83$.

a) Set up a linear system of equations that expresses the conditions described.

b) Solve this system of equations.

Exercise *

Kevin pays ${}2.50$€ for a winter bunch of flowers with ${}3$ snowdrops and ${}4$ mistletoes, and Jennifer pays ${}2.30$€ for a bunch with ${}5$ snowdrops and ${}2$ mistletoes. How much does a bunch with one snowdrop and ${}11$ mistletoes cost?

Exercise

We look at a clock with hour and minute hands. Now it is 6 o'clock, so that both hands have opposite directions. When will the hands have opposite directions again?

Exercise

Compute the following product of matrices

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

The ${}i$-th standard vector of length ${}n$ is the vector of length ${}n$ where there is ${}1$ at the ${}i$-th place, and ${}0$ everywhere else.

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 *

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

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.

Exercise

Let ${}2\times 2$-matrices ${}A={\begin{pmatrix}a_{11}&a_{12}\\a_{21}&a_{22}\end{pmatrix}}$ and ${}B={\begin{pmatrix}b_{11}&b_{12}\\b_{21}&b_{22}\end{pmatrix}}$ be given. The product ${}A\cdot B$ is usually computed by the multiplication rule "row x column“; for this, we have to perform altogether ${}8$ multiplications in the field ${}K$. We describe a procedure for the matrix multiplication, in which only ${}7$ multiplications (but more additions) are necessary. We set

${}m_{1}={\left(a_{11}+a_{22}\right)}\cdot {\left(b_{11}+b_{22}\right)}\,,$

${}m_{2}={\left(a_{21}+a_{22}\right)}\cdot b_{11}\,,$

${}m_{3}=a_{11}\cdot {\left(b_{12}-b_{22}\right)}\,,$

${}m_{4}=a_{22}\cdot {\left(b_{21}-b_{11}\right)}\,,$

${}m_{5}={\left(a_{11}+a_{12}\right)}\cdot b_{22}\,,$

${}m_{6}={\left(a_{21}-a_{11}\right)}\cdot {\left(b_{11}+b_{12}\right)}\,,$

${}m_{7}={\left(a_{12}-a_{22}\right)}\cdot {\left(b_{21}+b_{22}\right)}\,.$

Show that the coefficients of the product matrix

${}AB=C={\begin{pmatrix}c_{11}&c_{12}\\c_{21}&c_{22}\end{pmatrix}}\,$

satisfy the equations

${}c_{11}=m_{1}+m_{4}-m_{5}+m_{7}\,,$

${}c_{12}=m_{3}+m_{5}\,,$

${}c_{21}=m_{2}+m_{4}\,,$

${}c_{22}=m_{1}-m_{2}+m_{3}+m_{6}\,.$

For a matrix ${}M$, we denote by ${}M^{n}$ the ${}n$-fold matrix product of ${}M$ with itself. This is also 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

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 ${}MD$.

The main difficulty in the following exercise is to prove associativity for the multiplication (see Exercise 4.24 ) and the distributive law.

Exercise

Let ${}K$ be a field, and ${}n\in \mathbb {N} _{+}$. Show that the set of all square ${}n\times n$-matrices over ${}K$, with the addition of matrices and with the product of matrices as multiplication, forms a ring.

Exercise

Let ${}K$ be a field, and ${}m,n,p\in \mathbb {N}$. Prove that the transpose of a matrix satisfies the following properties (where ${}A,B\in \operatorname {Mat} _{m\times n}(K)$, ${}C\in \operatorname {Mat} _{n\times p}(K)$, and ${}s\in K$).

1. ${}{({A^{\text{tr}}})^{\text{tr}}}=A\,.$

2. ${}{(A+B)^{\text{tr}}}={A^{\text{tr}}}+{B^{\text{tr}}}\,.$

3. ${}{(sA)^{\text{tr}}}=s\cdot {A^{\text{tr}}}\,.$

4. ${}{(A\circ C)^{\text{tr}}}={C^{\text{tr}}}\circ {A^{\text{tr}}}\,.$

Hand-in-exercises

Exercise (3 marks)

Solve the linear system

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

over the field ${}K=\{0,1,2,3,4,5,6\}$ from Example 3.9 .

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,

${}M^{4}=MMMM=0\,.$

For the following statement, we will get in

[[Linear mapping/Matrix/Composition/Fact|]]

In the correspondence between linear mappings and matrices, the composition of linear mappings corresponds to the matrix multiplication. More precisely: let ${}U,V,W$ denote vector spaces over a field ${}K$ with bases

${\mathfrak {u}}=u_{1},\ldots ,u_{p},\,{\mathfrak {v}}=v_{1},\ldots ,v_{n}{\text{ and }}{\mathfrak {w}}=w_{1},\ldots ,w_{m}.$

Let

$\psi :U\longrightarrow V{\text{ and }}\varphi :V\longrightarrow W$

denote linear mappings. Then, for the describing matrix of ${}\psi ,\,\varphi$, and of the composition ${}\varphi \circ \psi$, the relation

${}M_{\mathfrak {w}}^{\mathfrak {u}}(\varphi \circ \psi )=(M_{\mathfrak {w}}^{\mathfrak {v}}(\varphi ))\circ (M_{\mathfrak {v}}^{\mathfrak {u}}(\psi ))\,$

holds.

Proof

We consider the commutative diagram

${\begin{matrix}K^{p}&{\stackrel {M_{\mathfrak {v}}^{\mathfrak {u}}(\psi )}{\longrightarrow }}&K^{n}&{\stackrel {M_{\mathfrak {w}}^{\mathfrak {v}}(\varphi )}{\longrightarrow }}&K^{m}&\\\!\!\!\!\!\Psi _{\mathfrak {u}}\downarrow &&\!\!\!\!\!\Psi _{\mathfrak {v}}\downarrow &&\downarrow \Psi _{\mathfrak {w}}\!\!\!\!\!&&\\U&{\stackrel {\psi }{\longrightarrow }}&V&{\stackrel {\varphi }{\longrightarrow }}&W&\!\!\!\!\!,\\\end{matrix}}$

where the commutativity rests on the identities

${}\varphi \circ \Psi _{\mathfrak {v}}=\Psi _{\mathfrak {w}}\circ M_{\mathfrak {w}}^{\mathfrak {v}}(\varphi )\,$

from Lemma 10.14 . The (inverse) coordinate mappings ${}\psi _{\mathfrak {v}}$ are bijective. Therefore, we have

${}M_{\mathfrak {w}}^{\mathfrak {v}}(\varphi )=\Psi _{\mathfrak {w}}^{-1}\circ \varphi \circ \Psi _{\mathfrak {v}}\,.$

Hence, we get altogether

${}{\begin{aligned}M_{\mathfrak {w}}^{\mathfrak {u}}(\varphi \circ \psi )&=\Psi _{\mathfrak {w}}^{-1}\circ (\varphi \circ \psi )\circ \Psi _{\mathfrak {u}}\\&={\left(\Psi _{\mathfrak {w}}^{-1}\circ \varphi \right)}\circ {\left(\Psi _{\mathfrak {v}}\circ \Psi _{\mathfrak {v}}^{-1}\right)}\circ {\left(\psi \circ \Psi _{\mathfrak {u}}\right)}\\&={\left(\Psi _{\mathfrak {w}}^{-1}\circ \varphi \circ \Psi _{\mathfrak {v}}\right)}\circ {\left(\Psi _{\mathfrak {v}}^{-1}\circ \psi \circ \Psi _{\mathfrak {u}}\right)}\\&=M_{\mathfrak {w}}^{\mathfrak {v}}(\varphi )\circ M_{\mathfrak {v}}^{\mathfrak {u}}(\psi ),\end{aligned}}$

where we have everywhere compositions of mappings. Due to Exercise 10.20 , the composition of mappings corresponds to the matrix multiplication.

$\Box$

a simpler proof via the relation between matrices and linear mappings.

Exercise (4 marks)

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 (4 marks)

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

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

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