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

Exercise for the break

Exercise

Linear system/2x+3y is 7 and 5x+4y is 3/Exercise

Exercises

Exercise

Linear system/3x is 5/Several fields/1/Exercise

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

Linear equation/(2+5i)z is (3-7i)/Modulus/Exercise

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?

Exercise

Flatbread/Thaler/Exercise

Exercise

Dating service/11 Minutes/Exercise

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, that 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 a ${}m\times n$- matrix. Show that the matrix product ${}Me_{j}$ of ${}M$ with the ${}j$-th standard vector (regarded as 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 parantheses.

Exercise

Strassen-algorithm/2x2/Exercise

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) and the distributivity law.

Exercise

Square matrices/Matrix ring/Exercise

Exercise

Let ${}K$ be a field and ${}m,n,p\in \mathbb {N}$. Prove that the transpose of a matrix satisfy 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 Linear system/Over Z mod 7/1/Exercise

Linear system/Over Z mod 7/1/Exercise

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 soon a simple 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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