# Kurs:Mathematik für Anwender (Osnabrück 2011-2012)/Teil I/Arbeitsblatt 8/en

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Warm-up-exercises

### Exercise

Let ${\displaystyle {}K}$ be a field and let ${\displaystyle {}V}$ be a ${\displaystyle {}K}$-vector space of dimension ${\displaystyle {}n=\operatorname {dim} _{}\left(V\right)}$. Suppose that ${\displaystyle {}n}$ vectors ${\displaystyle {}v_{1},\ldots ,v_{n}}$ in ${\displaystyle {}V}$ are given. Prove that the following facts are equivalent.

1. ${\displaystyle {}v_{1},\ldots ,v_{n}}$ form a basis for ${\displaystyle {}V}$.
2. ${\displaystyle {}v_{1},\ldots ,v_{n}}$ form a system of generators for ${\displaystyle {}V}$.
3. ${\displaystyle {}v_{1},\ldots ,v_{n}}$ are linearly independent.

### Exercise

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

### Exercise

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

### Exercise

Let ${\displaystyle {}K}$ be a field and let ${\displaystyle {}V}$ and ${\displaystyle {}W}$ be two finite-dimensional ${\displaystyle {}K}$-vector spaces with ${\displaystyle {}\operatorname {dim} _{}\left(V\right)=n}$ and ${\displaystyle {}\operatorname {dim} _{}\left(W\right)=m}$. What is the dimension of the Cartesian product ${\displaystyle {}V\times W}$?

### Exercise

Let ${\displaystyle {}V}$ be a finite-dimensional vector space over the complex numbers, and let ${\displaystyle {}v_{1},\ldots ,v_{n}}$ be a basis of ${\displaystyle {}V}$. Prove that the family of vectors

${\displaystyle v_{1},\ldots ,v_{n}{\text{ and }}iv_{1},\ldots ,iv_{n}}$

form a basis for ${\displaystyle {}V}$, considered as a real vector space.

### Exercise

Consider the standard basis ${\displaystyle {}e_{1},e_{2},e_{3},e_{4}}$ in ${\displaystyle {}\mathbb {R} ^{4}}$ and the three vectors

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

Prove that these vectors are linearly independent and extend them to a basis by adding an appropriate standard vector as shown in Theorem 8.2. Can one take any standard vector?

### Exercise

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

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

### Exercise

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

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

### Exercise

We consider the families of vectors

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

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

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

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

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

Hand-in-exercises

### Exercise (4 points)

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

### Exercise (3 points)

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

${\displaystyle U=\langle v_{i},\,i=1,\ldots ,m\rangle }$
be the subspace they span. Prove that the family is linearly independent if and only if the dimension of ${\displaystyle {}U}$ is exactly ${\displaystyle {}m}$.

### Exercise (4 points)

Determine the transformation matrices ${\displaystyle {}M_{\mathfrak {v}}^{\mathfrak {u}}}$ and ${\displaystyle {}M_{\mathfrak {u}}^{\mathfrak {v}}}$ for the standard basis ${\displaystyle {}{\mathfrak {u}}}$ and the basis ${\displaystyle {}{\mathfrak {v}}}$ of ${\displaystyle {}\mathbb {R} ^{3}}$ which is given by the vectors

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

### Exercise (6 points)

We consider the families of vectors

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

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

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

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

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