# Mathematics for Applied Sciences (Osnabrück 2011-2012)/Part I/Exercise sheet 6

    Warm-up-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 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 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.
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 $K$ 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 $(AB)C=A(BC)$.
For a matrix $M$ we denote by $M^{n}$ the $n$-th 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 $K$ 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 $K$ 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 $K$ 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

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

Let $K$ be a field, and let $V$ be a $K$-vector space. Let $U,W\subseteq V$ be subspaces of $V$. Prove that the union $U\cup W$ is a subspace of $V$ if and only if $U\subseteq W$ or $W\subseteq U$.
    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 (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 $s\in K$).
* We have $0v=0$.
* We have $s0=0$.
* We have $(-1)v=-v$.
* 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 $V$ which satisfy two of the subspace axioms, but not the third.