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

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\zwischenueberschrift{Warm-up-exercises}




\inputexercise
{}
{

Let $K$ be a field and let $V$ be a $K$-vector space of dimension
\mathl{n= \operatorname{dim}_{ } { \left( V \right) }}{.} Suppose that $n$ vectors
\mathl{v_1 , \ldots , v_n}{} in $V$ are given. Prove that the following facts are equivalent. \aufzaehlungdrei{
\mathl{v_1 , \ldots , v_n}{} form a basis for $V$. }{
\mathl{v_1 , \ldots , v_n}{} form a system of generators for $V$. }{
\mathl{v_1 , \ldots , v_n}{} are linearly independent. }

}
{} {}




\inputexercise
{}
{

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

}
{} {}




\inputexercise
{}
{

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

}
{} {}




\inputexercise
{}
{

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

}
{} {}




\inputexercise
{}
{

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
\mathdisp {v_1 , \ldots , v_n \text{ and } iv_1 , \ldots , iv_n} { }
form a basis for $V$, considered as a real vector space.

}
{} {}




\inputexercise
{}
{

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

}
{} {}




\inputexercise
{}
{

Determine the \definitionsverweis {transformation matrices}{}{} \mathkor {} {M^{ \mathfrak{ u } }_{ \mathfrak{ v } }} {and} {M^{ \mathfrak{ v } }_{ \mathfrak{ u } }} {} for the \definitionsverweis {standard basis}{}{} $\mathfrak{ u }$ and the basis $\mathfrak{ v }$ in $\R^4$ which is given by \mathlistdisp {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}} {and} {v_4 = \begin{pmatrix} 0 \\1\\ 0\\0 \end{pmatrix}} {.}

}
{} {}




\inputexercise
{}
{

Determine the \definitionsverweis {transformation matrices}{}{} \mathkor {} {M^{ \mathfrak{ u } }_{ \mathfrak{ v } }} {and} {M^{ \mathfrak{ v } }_{ \mathfrak{ u } }} {} for the \definitionsverweis {standard basis}{}{} $\mathfrak{ u }$ and the basis $\mathfrak{ v }$ of ${\mathbb C}^2$ which is given by the vectors \mathlistdisp {v_1 = \begin{pmatrix} 3+5i \\1-i \end{pmatrix}} {and} {v_2 = \begin{pmatrix} 2+3i \\4+i \end{pmatrix}} {} {} {.}

}
{} {}




\inputexercise
{}
{

We consider the families of vectors
\mathdisp {\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 $\R^2$.

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

b) Let
\mathl{P \in \R^2}{} denote the point which has the coordinates
\mathl{(-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 \definitionsverweis {transformation matrix}{}{} which describes the \definitionsverweis {change of basis}{}{} from $\mathfrak{ v }$ to $\mathfrak{ u }$.

}
{} {}






\zwischenueberschrift{Hand-in-exercises}




\inputexercise
{4}
{

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

}
{} {}




\inputexercise
{3}
{

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
\mathdisp {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 $U$ is exactly $m$.

}
{} {}




\inputexercise
{4}
{

Determine the \definitionsverweis {transformation matrices}{}{} \mathkor {} {M^{ \mathfrak{ u } }_{ \mathfrak{ v } }} {and} {M^{ \mathfrak{ v } }_{ \mathfrak{ u } }} {} for the \definitionsverweis {standard basis}{}{} $\mathfrak{ u }$ and the basis $\mathfrak{ v }$ of $\R^3$ which is given by the vectors \mathlistdisp {v_1 = \begin{pmatrix} 4 \\5\\ 1 \end{pmatrix}} {} {v_2 = \begin{pmatrix} 2 \\3\\ -8 \end{pmatrix}} {and} {v_3 = \begin{pmatrix} 5 \\7\\ -3 \end{pmatrix}} {.}

}
{} {}




\inputexercise
{6}
{

We consider the families of vectors
\mathdisp {\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 $\R^3$.

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

b) Let
\mathl{P \in \R^3}{} denote the point which has the coordinates
\mathl{(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 \definitionsverweis {transformation matrix}{}{} which describes the \definitionsverweis {change of basis}{}{} from $\mathfrak{ v }$ to $\mathfrak{ u }$.

}
{} {}



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