Mathematics for Applied Sciences (Osnabrück 2011-2012)/Part I/Exercise sheet 10

Warm-up-exercises

Exercise

The telephone companies ${}A,B$ and ${}C$ compete for a market, where the market customers in a year ${}j$ are given by the customers-tuple ${}K_{j}=(a_{j},b_{j},c_{j})$ (where ${}a_{j}$ is the number of customers of ${}A$ in the year ${}j$ etc.). There are customers passing from one provider to another one during a year.

1. The customers of ${}A$ remain for ${}80\%$ with ${}A$, while ${}10\%$ of them goes to ${}B$, and the same percentage goes to ${}C$.
2. The customers of ${}B$ remain for ${}70\%$ with ${}B$, while ${}10\%$ of them goes to ${}A$, and ${}20\%$ goes to ${}C$.
3. The customers of ${}C$ remain for ${}50\%$ with ${}C$, while ${}20\%$ of them goes to ${}A$, and ${}30\%$ goes to ${}B$.

a) Determine the linear map (i.e. the matrix) that expresses the customers-tuple ${}K_{j+1}$ with respect to ${}K_{j}$.

b) Which customers-tuple arises from the customers-tuple ${}(12000,10000,8000)$ within one year?

c) Which customers-tuple arises from the customers-tuple ${}(10000,0,0)$ in four years?

Exercise *

Let ${}K$ be a field and let ${}V$ and ${}W$ be vector spaces over ${}K$ of dimensions ${}n$ and ${}m$. Let

$\varphi \colon V\longrightarrow W$

be a linear map, described by the matrix ${}M\in \operatorname {Mat} _{m\times n}(K)$ with respect to two bases. Prove that ${}\varphi$ is surjective if and only if the columns of the matrix form a system of generators for ${}K^{m}$.

Exercise

Let ${}K$ be a field, and let ${}V$ and ${}W$ be ${}K$-vector spaces. Let

$\varphi \colon V\longrightarrow W$

be a bijective linear map. Prove that also the inverse map

$\varphi ^{-1}\colon W\longrightarrow V$

is linear.

Exercise

Determine the inverse matrix of

${}M={\begin{pmatrix}2&7\\-4&9\end{pmatrix}}\,$

Exercise

Determine the inverse matrix of

${}M={\begin{pmatrix}1&2&3\\6&-1&-2\\0&3&7\end{pmatrix}}\,.$

Exercise

Determine the inverse matrix of the complex matrix

${}M={\begin{pmatrix}2+3{\mathrm {i} }&1-{\mathrm {i} }\\5-4{\mathrm {i} }&6-2{\mathrm {i} }\end{pmatrix}}\,.$

Exercise *

a) Determine if the complex matrix

${}M={\begin{pmatrix}2+5{\mathrm {i} }&1-2{\mathrm {i} }\\3-4{\mathrm {i} }&6-2{\mathrm {i} }\end{pmatrix}}\,$

is invertible.

b) Find a solution to the inhomogeneous linear system of equations

${}M{\begin{pmatrix}z_{1}\\z_{2}\end{pmatrix}}={\begin{pmatrix}54+72{\mathrm {i} }\\0\end{pmatrix}}\,.$

Exercise

Prove that the matrix

${\begin{pmatrix}0&0&k+2&k+1\\0&0&k+1&k\\-k&k+1&0&0\\k+1&-(k+2)&0&0\end{pmatrix}}$

for all ${}k$ is the inverse of itself.

Exercise

We consider the linear map

$\varphi \colon K^{3}\longrightarrow K^{2},{\begin{pmatrix}x\\y\\z\end{pmatrix}}\longmapsto {\begin{pmatrix}1&2&5\\4&1&1\end{pmatrix}}{\begin{pmatrix}x\\y\\z\end{pmatrix}}.$

Let ${}U\subseteq K^{3}$ be the subspace of ${}K^{3}$, defined by the linear equation ${}2x+3y+4z=0$, and let ${}\psi$ be the restriction of ${}\varphi$ on ${}U$. On ${}U$, there are given vectors of the form

$u=(0,1,a),\,v=(1,0,b){\text{ and }}w=(1,c,0).$

Compute the "change of basis" matrix between the bases

${\mathfrak {b}}_{1}=v,w,\,{\mathfrak {b}}_{2}=u,w{\text{ and }}{\mathfrak {b}}_{3}=u,v$

of ${}U$, and the transformation matrix of ${}\psi$ with respect to these three bases (and the standard basis of ${}K^{2}$).

Exercise

Prove that the elementary matrices are invertible. What are the inverse matrices of the elementary matrices?

Exercise

Let ${}K$ be a field and ${}M$ a ${}n\times n$-matrix with entries in ${}K$. Prove that the multiplication by the elementary matrices from the left with M has the following effects.

1. ${}V_{ij}\circ M=$ exchange of the ${}i$-th and the ${}j$-th row of ${}M$.
2. ${}(S_{k}(s))\circ M=$ multiplication of the ${}k$-th row of ${}M$ by ${}s$.
3. ${}(A_{ij}(a))\circ M=$ addition of ${}a$-times the ${}j$-th row of ${}M$ to the ${}i$-th row (${}i\neq j$).

Exercise

Describe what happens when a matrix is multiplied from the right by an elementary matrix.

Hand-in-exercises

Exercise (3 marks)

Compute the inverse matrix of

${}M={\begin{pmatrix}2&3&2\\5&0&4\\1&-2&3\end{pmatrix}}\,.$

Exercise (3 marks)

Perform the procedure to find the inverse matrix of the matrix

${\begin{pmatrix}a&b\\c&d\end{pmatrix}}$

under the assumption that ${}ad-bc\neq 0$.

Exercise (6 (3+1+2) marks)

An animal population consists of babies (first year), freshers (second year), rockers (third year), mature ones (fourth year), and veterans (fifth year), these animals can not become older. The total stock of these animals in a given year ${}j$ is given by a ${}5$-tuple ${}B_{j}=(b_{1,j},b_{2,j},b_{3,j},b_{4,j},b_{5,j})$.

During a year, ${}7/8$ of the babies become freshers, ${}9/10$ of the freshers become rockers, ${}5/6$ of the rockers become mature ones, and ${}2/3$ of the mature ones reach the fifth year.

Babies and freshers can not reproduce yet, then they reach sexual maturity, and ${}10$ rockers generate ${}5$ new pets, and ${}10$ of the mature ones generate ${}8$ new babies, and the babies are born one year later.

a) Determine the linear map (i.e., the matrix) that expresses the total stock ${}B_{j+1}$ with respect to the stock ${}B_{j}$.

b) What will happen to the stock ${}(200,150,100,100,50)$ in the next year?

c) What will happen to the stock ${}(0,0,100,0,0)$ in five years?

Exercise (3 marks)

Let ${}z\in \mathbb {C}$ be a complex number and let

$\mathbb {C} \longrightarrow \mathbb {C} ,w\longmapsto zw,$

be the multiplication map, which is a ${}\mathbb {C}$-linear map. How does the matrix related to this map with respect to the real basis ${}1$ and ${}{\mathrm {i} }$ look like? Let ${}z_{1}$ and ${}z_{2}$ be complex numbers with corresponding real matrices ${}M_{1}$ and ${}M_{2}$. Prove that the matrix product ${}M_{2}\circ M_{1}$ is the real matrix corresponding to ${}z_{1}z_{2}$.