# Kurs:Mathematik für Anwender (Osnabrück 2011-2012)/Teil I/Arbeitsblatt 6/en

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Warm-up-exercises

### Exercise

Compute the following product of matrices

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

### Exercise

Compute over the complex numbers the following product of matrices

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

### Exercise

Determine the product of matrices

${\displaystyle e_{i}\circ e_{j},}$
where the ${\displaystyle {}i}$-th standard vector

(of length ${\displaystyle {}n}$) is considered as a row vector and the ${\displaystyle {}j}$-th standard vector (also of length ${\displaystyle {}n}$) is considered as a column vector.

### Exercise

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

### Exercise

Compute the product of matrices

${\displaystyle {\begin{pmatrix}2+{\mathrm {i} }&1-{\frac {1}{2}}{\mathrm {i} }&4{\mathrm {i} }\\-5+7{\mathrm {i} }&{\sqrt {2}}+{\mathrm {i} }&0\end{pmatrix}}{\begin{pmatrix}-5+4{\mathrm {i} }&3-2{\mathrm {i} }\\{\sqrt {2}}-{\mathrm {i} }&e+\pi {\mathrm {i} }\\1&-{\mathrm {i} }\end{pmatrix}}{\begin{pmatrix}1+{\mathrm {i} }\\2-3{\mathrm {i} }\end{pmatrix}}}$
according to the two possible parantheses.

For the following statements we will soon give a simple proof using the relationship between matrices and linear maps.

### Exercise

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

For a matrix ${\displaystyle {}M}$ we denote by ${\displaystyle {}M^{n}}$ the ${\displaystyle {}n}$-th product of ${\displaystyle {}M}$ with iself. This is also called the ${\displaystyle {}n}$-th power of the matrix.

### Exercise

Compute for the matrix

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

the powers

${\displaystyle M^{i},\,i=1,\ldots ,4.}$

### Exercise

Let ${\displaystyle K}$ be a field and let ${\displaystyle V}$ and ${\displaystyle W}$ be two vector spaces over ${\displaystyle K}$. Show that the product

${\displaystyle V\times W}$
is also a ${\displaystyle K}$-vector space.

### Exercise

Let ${\displaystyle {}K}$ be a field and ${\displaystyle {}I}$ an index set. Show that

${\displaystyle K^{I}:=\operatorname {Maps} \,(I,K)}$
with pointwise addition and scalar multiplication is a ${\displaystyle {}K}$-vector space.

### Exercise

Let ${\displaystyle {}K}$ be a field and let

${\displaystyle {\begin{matrix}a_{11}x_{1}+a_{12}x_{2}+\cdots +a_{1n}x_{n}&=&0\\a_{21}x_{1}+a_{22}x_{2}+\cdots +a_{2n}x_{n}&=&0\\\vdots &\vdots &\vdots \\a_{m1}x_{1}+a_{m2}x_{2}+\cdots +a_{mn}x_{n}&=&0\end{matrix}}}$
be a system of linear equations over ${\displaystyle {}K}$. Show that the set of all solutions of the system is a subspace of ${\displaystyle {}K^{n}}$. How is this solution space related to the solution spaces of the individual equations?

### Exercise

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

### Exercise

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

Hand-in-exercises

### Exercise (3 points)

Compute over the complex numbers the following product of matrices

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

### Exercise (4 points)

We consider the matrix

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

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

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

### Exercise (3 points)

Let ${\displaystyle {}K}$ be a field and let ${\displaystyle {}V}$ be a ${\displaystyle {}K}$-vector space. Show that the following properties hold (where ${\displaystyle \lambda \in K}$ and ${\displaystyle v\in V}$).

1. We have ${\displaystyle {}0v=0}$.
2. We have ${\displaystyle {}\lambda 0=0}$.
3. We have${\displaystyle {}(-1)v=-v}$.
4. If ${\displaystyle {}\lambda \neq 0}$ and ${\displaystyle {}v\neq 0}$ then ${\displaystyle {}\lambda v\neq 0}$.

### Exercise (3 points)

Give an example of a vector space ${\displaystyle {}V}$ and of three subsets of ${\displaystyle {}V}$ which satisfy two of the subspace axioms, but not the third.