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 ${}v_{1},\ldots ,v_{n}$ in ${}V$ are given. Prove that the following facts are equivalent.

1. ${}v_{1},\ldots ,v_{n}$ form a basis for ${}V$.
2. ${}v_{1},\ldots ,v_{n}$ form a system of generators for ${}V$.
3. ${}v_{1},\ldots ,v_{n}$ 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 ${}\leq d$ is a finite-dimensional linear subspace of ${}K[X]$. What is its dimension?

Exercise

Show that the set of real polynomials of degree ${}\leq 4$ which have a zero at ${}-2$ and a zero at ${}3$ is a finite dimensional subspace of ${}\mathbb {R} [X]$. 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 ${}v_{1},\ldots ,v_{n}$ be a basis of ${}V$. Prove that the family of vectors

$v_{1},\ldots ,v_{n}{\text{ and }}{\mathrm {i} }v_{1},\ldots ,{\mathrm {i} }v_{n}$

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 ${}\mathbb {R} ^{4}$ and the three vectors

${\begin{pmatrix}1\\3\\0\\-4\end{pmatrix}},\,{\begin{pmatrix}2\\1\\5\\7\end{pmatrix}}{\text{ and }}{\begin{pmatrix}-4\\9\\-5\\1\end{pmatrix}}.$

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 ${}M_{\mathfrak {v}}^{\mathfrak {u}}$ and ${}M_{\mathfrak {u}}^{\mathfrak {v}}$, for the standard basis ${}{\mathfrak {u}}$, and the basis ${}{\mathfrak {v}}$ in ${}\mathbb {R} ^{4}$, which is given by

$v_{1}={\begin{pmatrix}0\\0\\1\\0\end{pmatrix}},\,v_{2}={\begin{pmatrix}1\\0\\0\\0\end{pmatrix}},\,v_{3}={\begin{pmatrix}0\\0\\0\\1\end{pmatrix}},\,v_{4}={\begin{pmatrix}0\\1\\0\\0\end{pmatrix}}.$

Exercise

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

$v_{1}={\begin{pmatrix}3+5{\mathrm {i} }\\1-{\mathrm {i} }\end{pmatrix}}{\text{ and }}v_{2}={\begin{pmatrix}2+3{\mathrm {i} }\\4+{\mathrm {i} }\end{pmatrix}}.$

Exercise

We consider the families of vectors

${\mathfrak {v}}={\begin{pmatrix}7\\-4\end{pmatrix}},\,{\begin{pmatrix}8\\1\end{pmatrix}}\,\,{\text{ and }}\,\,{\mathfrak {u}}={\begin{pmatrix}4\\6\end{pmatrix}},\,{\begin{pmatrix}7\\3\end{pmatrix}}$

in ${}\mathbb {R} ^{2}$.

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

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

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

Hand-in-exercises

Exercise (4 marks)

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

Exercise (2 marks)

Let ${}K$ be a field, and let ${}V$ be a ${}K$-vector space. Let ${}v_{1},\ldots ,v_{m}$ be a family of vectors in ${}V$, and let

${}U=\langle v_{i},\,i=1,\ldots ,m\rangle \,$

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 ${}M_{\mathfrak {v}}^{\mathfrak {u}}$ and ${}M_{\mathfrak {u}}^{\mathfrak {v}}$ for the standard basis ${}{\mathfrak {u}}$ and the basis ${}{\mathfrak {v}}$ of ${}\mathbb {R} ^{3}$ that is given by the vectors

$v_{1}={\begin{pmatrix}4\\5\\1\end{pmatrix}},\,\,v_{2}={\begin{pmatrix}2\\3\\-8\end{pmatrix}},\,{\text{ and }}\,v_{3}={\begin{pmatrix}5\\7\\-3\end{pmatrix}}$

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

We consider the families of vectors

${\mathfrak {v}}={\begin{pmatrix}1\\2\\3\end{pmatrix}},\,{\begin{pmatrix}4\\7\\1\end{pmatrix}},\,{\begin{pmatrix}0\\2\\5\end{pmatrix}}\,\,{\text{ and }}\,\,{\mathfrak {u}}={\begin{pmatrix}0\\2\\4\end{pmatrix}},\,{\begin{pmatrix}6\\6\\1\end{pmatrix}},\,{\begin{pmatrix}3\\5\\-2\end{pmatrix}}$

in ${}\mathbb {R} ^{3}$.

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

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

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