Mathematics for Applied Sciences (Osnabrück 2011-2012)/Part I/Exercise sheet 8

Warm-up-exercises

Exercise

Let $K$ be a field, and let $V$ be a $K$-vector space of dimension $n={\dim }_K\left(V\right)$. Suppose that $n$ vectors $v_1,\dots ,v_n$ in $V$ are given. Prove that the following facts are equivalent.

1. $v_1,\dots ,v_n$ form a basis for $V$.
2. $v_1,\dots ,v_n$ form a system of generators for $V$.
3. $v_1,\dots ,v_n$ are linearly independent.

Exercise

Let $K$ be a field, and let $K[X]$ denote the polynomial ring over $K$. Let $d\in \mathbb{N}$. Show that the set of all polynomials of degree $\le d$ is a finite-dimensional linear subspace of $K[X]$. What is its dimension?

Exercise

Show that the set of real polynomials of degree $\le 4$ which have a zero at $-2$ and a zero at $3$ is a finite dimensional subspace of $ℝ[X]$. Determine the dimension of this vector space.

Exercise *

Let $K$ be a field, and let $V$ and $W$ be two finite-dimensional $K$-vector spaces with

${\dim }_K\left(V\right)=n$

and

${\dim }_K\left(W\right)=m.$

What is the dimension of the Cartesian product $V\times W$?

Exercise

Let $V$ be a finite-dimensional vector space over the complex numbers, and let $v_1,\dots ,v_n$ be a basis of $V$. Prove that the family of vectors

$v_1,\dots ,v_n\text{ and }\mathrm{i}{v}_1,\dots ,\mathrm{i}{v}_n$

forms a basis for $V$, considered as a real vector space.

Exercise

Consider the standard basis $e_1,e_2,e_3,e_4$ in ${ℝ}^4$ and the three vectors

$\left(\begin{array}{c}1\\ 3\\ 0\\ -4\end{array}\right),\left(\begin{array}{c}2\\ 1\\ 5\\ 7\end{array}\right)\text{ and }\left(\begin{array}{c}-4\\ 9\\ -5\\ 1\end{array}\right).$

Prove that these vectors are linearly independent, and extend them to a basis by adding an appropriate standard vector, as shown in the base exchange theorem. Can one take any standard vector?

Exercise

Determine the transformation matrices $M_{𝔳}^{𝔲}$ and $M_{𝔲}^{𝔳}$, for the standard basis $𝔲$, and the basis $𝔳$ in ${ℝ}^4$, which is given by

$v_1=\left(\begin{array}{c}0\\ 0\\ 1\\ 0\end{array}\right),v_2=\left(\begin{array}{c}1\\ 0\\ 0\\ 0\end{array}\right),v_3=\left(\begin{array}{c}0\\ 0\\ 0\\ 1\end{array}\right),v_4=\left(\begin{array}{c}0\\ 1\\ 0\\ 0\end{array}\right).$

Exercise

Determine the transformation matrices $M_{𝔳}^{𝔲}$ and $M_{𝔲}^{𝔳}$ for the standard basis $𝔲$ and the basis $𝔳$ of ${ℂ}^2$ that is given by the vectors

$v_1=\left(\begin{array}{c}3+5\mathrm{i}\\ 1-\mathrm{i}\end{array}\right)\text{ and }{v}_2=\left(\begin{array}{c}2+3\mathrm{i}\\ 4+\mathrm{i}\end{array}\right).$

Exercise

We consider the families of vectors

$𝔳=\left(\begin{array}{c}7\\ -4\end{array}\right),\left(\begin{array}{c}8\\ 1\end{array}\right)\text{ and }𝔲=\left(\begin{array}{c}4\\ 6\end{array}\right),\left(\begin{array}{c}7\\ 3\end{array}\right)$

in ${ℝ}^2$.

a) Show that $𝔳$ and $𝔲$ are both a basis of ${ℝ}^2$.

b) Let $P\in {ℝ}^2$ denote the point that has the coordinates $(-2,5)$ with respect to the basis $𝔳$. What are the coordinates of this point with respect to the basis $𝔲$?

c) Determine the transformation matrix that describes the change of bases from $𝔳$ to $𝔲$.

Hand-in-exercises

Exercise (4 marks)

Show that the set of all real polynomials of degree $\le 6$ that have a zero at $-1$, at $0$ and at $1$, is a finite-dimensional subspace of $ℝ[X]$. Determine the dimension of this vector space.

Exercise (2 marks)

Let $K$ be a field, and let $V$ be a $K$-vector space. Let $v_1,\dots ,v_m$ be a family of vectors in $V$, and let

$U=⟨v_i,i=1,\dots ,m⟩$

be the linear subspace they span. Prove that the family is linearly independent if and only if the dimension of $U$ is exactly $m$.

Exercise (4 marks)

Determine the transformation matrices $M_{𝔳}^{𝔲}$ and $M_{𝔲}^{𝔳}$ for the standard basis $𝔲$ and the basis $𝔳$ of ${ℝ}^3$ that is given by the vectors

$v_1=\left(\begin{array}{c}4\\ 5\\ 1\end{array}\right),v_2=\left(\begin{array}{c}2\\ 3\\ -8\end{array}\right),\text{ and }{v}_3=\left(\begin{array}{c}5\\ 7\\ -3\end{array}\right)$

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

We consider the families of vectors

$𝔳=\left(\begin{array}{c}1\\ 2\\ 3\end{array}\right),\left(\begin{array}{c}4\\ 7\\ 1\end{array}\right),\left(\begin{array}{c}0\\ 2\\ 5\end{array}\right)\text{ and }𝔲=\left(\begin{array}{c}0\\ 2\\ 4\end{array}\right),\left(\begin{array}{c}6\\ 6\\ 1\end{array}\right),\left(\begin{array}{c}3\\ 5\\ -2\end{array}\right)$

in ${ℝ}^3$.

a) Show that $𝔳$ and $𝔲$ are both a basis of ${ℝ}^3$.

b) Let $P\in {ℝ}^3$ denote the point that has the coordinates $(2,5,4)$ with respect to the basis $𝔳$. What are the coordinates of this point with respect to the basis $𝔲$?

c) Determine the transformation matrix that describes the change of basis from $𝔳$ to $𝔲$.