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 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 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 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 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 ${}W\subseteq U$.

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 ${}s\in K$).

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.