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

Other exercises

Exercise

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

$\varphi \colon V\longrightarrow W$

be a linear map. Prove that the graph of the map is a subspace of the Cartesian product ${}V\times W$.

Exercise

Let ${}K$ be a field and let ${}V$ be a ${}K$-vector space. Prove that for ${}v\in V$ the map

$K\longrightarrow V,\lambda \longmapsto \lambda v,$

is linear.

Exercise

How does the graph of a linear map

$f\colon \mathbb {R} \longrightarrow \mathbb {R} ,$

$g\colon \mathbb {R} \longrightarrow \mathbb {R} ^{2},$

$h\colon \mathbb {R} ^{2}\longrightarrow \mathbb {R}$

look like? How can you see in a sketch of the graph the kernel of the map?

Exercise

Let ${}K$ be a field and let ${}V$ and ${}W$ be ${}K$-vector spaces. Let ${}v_{1},\ldots ,v_{n}$ be a system of generators for ${}V$ and let ${}w_{1},\ldots ,w_{n}$ be a family of vectors in ${}W$.

a) Prove that there is at most one linear map

$\varphi \colon V\longrightarrow W$

such that ${}\varphi (v_{i})=w_{i}$ for all ${}i$.

b) Give an example of such a situation, where there is no linear mapping with ${}\varphi (v_{i})=w_{i}$ for all ${}i$.

Exercise

Let ${}K$ be a field and let ${}A$ be a ${}m\times n$-matrix and ${}B$ a ${}n\times p$-matrix over ${}K$. Prove the following relationships concerning the rank

$\operatorname {rk} \,AB\leq \operatorname {rk} \,A{\text{ and }}\operatorname {rk} \,AB\leq \operatorname {rk} \,B.$

Prove that equality on the left occurs if ${}B$ is invertible, and equality on the right occurs if ${}A$ is invertible. Give an example of non-invertible matrices ${}A$ and ${}B$ such that equality on the left and on the right occurs.

Exercise

Prove that the series

$\sum _{n=1}^{\infty }{\frac {1}{n(n+1)}}$

converges with sum equal to ${}1$.

Exercise

Examine for each of the following subsets ${}M\subseteq \mathbb {Q}$ the concepts upper bound, lower bound, supremum, infimum, maximum and minimum.

1. ${}\{2,-3,-4,5,6,-1,1\}$,
2. ${}\left\{{\frac {1}{2}},{\frac {-3}{7}},{\frac {-4}{9}},{\frac {5}{9}},{\frac {6}{13}},{\frac {-1}{3}},{\frac {1}{4}}\right\}$,
3. ${}]-5,2]$,
4. ${}{\left\{{\frac {1}{n}}\mid n\in \mathbb {N} _{+}\right\}}$,
5. ${}{\left\{{\frac {1}{n}}\mid n\in \mathbb {N} _{+}\right\}}\cup \{0\}$,
6. ${}\mathbb {Q} _{-}$,
7. ${}{\left\{x\in \mathbb {Q} \mid x^{2}\leq 2\right\}}$,
8. ${}{\left\{x\in \mathbb {Q} \mid x^{2}\leq 4\right\}}$,
9. ${}{\left\{x^{2}\mid x\in \mathbb {Z} \right\}}$.

Exercise

Explain why the factorial function is continuous.