Kurs:Mathematik für Anwender (Osnabrück 2011-2012)/Teil I/Arbeitsblatt 7/en/latex

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\setcounter{section}{7}






\zwischenueberschrift{Warm-up-exercises}




\inputexercise
{}
{

Write in $\Q^2$ the vector
\mathdisp {(2,-7)} { }
as a linear combination of the vectors
\mathdisp {(5,-3) \text{ and } (-11,4)} { . }

}
{} {}




\inputexercise
{}
{

Write in ${\mathbb C}^2$ the vector
\mathdisp {(1,0)} { }
as a \definitionsverweis {linear combination}{}{} of the vectors
\mathdisp {(3+5i,-3+2i) \text{ and } (1-6i,4-i)} { . }

}
{} {}




\inputexercise
{}
{

Let $K$ be a field and let $V$ be a $K$-vector space. Let
\mathbed {v_i} {}
{i \in I} {}
{} {} {} {,} be a family of vectors in $V$ and let $w \in V$ be another vector. Assume that the family
\mathdisp {w, v_i, i \in I} { , }
is a system of generators of $V$ and that $w$ is a linear combination of the
\mathbed {v_i} {}
{i \in I} {}
{} {} {} {.} Prove that also
\mathbed {v_i} {}
{i \in I} {}
{} {} {} {,} is a system of generators of $V$.

}
{} {}




\inputexercise
{}
{ Vektorraum/Erzeugendensystem und aufgespannter Unterraum/Fakt/Beweis/Aufgabe/en }
{} {}




\inputexercise
{}
{

Show that the three vectors
\mathdisp {\begin{pmatrix} 0 \\1\\ 2\\1 \end{pmatrix},\, \begin{pmatrix} 4 \\3\\ 0\\2 \end{pmatrix},\, \begin{pmatrix} 1 \\7\\ 0\\-1 \end{pmatrix}} { }
in $\R^4$ are \definitionsverweis {linearly independent}{}{.}

}
{} {}




\inputexercise
{}
{

Give an example of three vectors in $\R^3$ such that each couple of them is linearly independent, but all three vectors are linearly dependent.

}
{} {}




\inputexercise
{}
{

Let $K$ be a field, let $V$ be a $K$-vector space and let
\mathbed {v_{ i }} {}
{i \in I} {}
{} {} {} {,} be a family of vectors in $V$. Prove the following facts. \aufzaehlungsechs{If the family is linearly independent then for each subset
\mathl{J \subseteq I}{} also the family
\mathbed {v_i} {,}
{i \in J} {}
{} {} {} {} is linearly independent. }{The empty family is linearly independent. }{If the family contains the null vector then it is not linearly independent. }{If a vector appears several times in the family, then the family is not linearly independent. }{A vector $v$ is linearly independent if and only if
\mathl{v \neq 0}{.} }{Two vectors \mathkor {} {v} {und} {u} {} are linearly independent if and only if $u$ is not a scalar multiple of $v$ and vice versa. }

}
{} {}




\inputexercise
{}
{

Let $K$ be a field, let $V$ be a $K$-vector space and let
\mathbed {v_{ i }} {}
{i \in I} {}
{} {} {} {,} be a family of vectors in $V$. Let
\mathbed {\lambda_i} {}
{i \in I} {}
{} {} {} {} be a family of elements $\neq 0$ in $K$. Prove that the family
\mathbed {v_i} {}
{i \in I} {}
{} {} {} {,} is linearly independent \zusatzklammer {a system of generators of $V$, a basis of $V$} {} {} if and only if the same holds for the family
\mathbed {\lambda_i v_i} {}
{i \in I} {}
{} {} {} {.}

}
{} {}




\inputexercise
{}
{

Determine a basis for the solution space of the linear equation
\mathdisp {3x+4y-2z+5w = 0} { . }

}
{} {}




\inputexercise
{}
{

Determine a basis for the solution space of the linear system of equations
\mathdisp {-2x+3y-z+4w = 0 \text{ and } 3z-2w =0} { . }

}
{} {}




\inputexercise
{}
{

Prove that in $\R^3$ the three vectors
\mathdisp {\begin{pmatrix} 2 \\1\\ 5 \end{pmatrix} \, , \begin{pmatrix} 1 \\3\\ 7 \end{pmatrix} \, , \begin{pmatrix} 4 \\1\\ 2 \end{pmatrix}} { }
are a basis.

}
{} {}




\inputexercise
{}
{

Establish if in ${\mathbb C}^2$ the two vectors
\mathdisp {\begin{pmatrix} 2+7i \\3-i \end{pmatrix} \text{ and } \begin{pmatrix} 15+26i \\13-7i \end{pmatrix}} { }
form a basis.

}
{} {}




\inputexercise
{}
{

Let $K$ be a field. Find a linear system of equations in three variables, whose solution space is exactly
\mathdisp {{ \left\{ \lambda \begin{pmatrix} 3 \\2\\ -5 \end{pmatrix} \mid \lambda \in K \right\} }} { . }

}
{} {}






\zwischenueberschrift{Hand-in-exercises}




\inputexercise
{3}
{

Write in $\Q^3$ the vector
\mathdisp {(2,5,-3)} { }
as a linear combination of the vectors
\mathdisp {(1,2,3), (0,1,1) \text{ und } (-1,2,4)} { . }
Prove that it cannot be expressed as a linear combination of two of the three vectors.

}
{} {}




\inputexercise
{2}
{

Establish if in $\R^3$ the three vectors
\mathdisp {\begin{pmatrix} 2 \\3\\ -5 \end{pmatrix} \, , \begin{pmatrix} 9 \\2\\ 6 \end{pmatrix} \, ,\begin{pmatrix} -1 \\4\\ -1 \end{pmatrix}} { }
form a basis.

}
{} {}




\inputexercise
{2}
{

Establish if in ${\mathbb C}^2$ the two vectors
\mathdisp {\begin{pmatrix} 2-7i \\-3+2i \end{pmatrix} \text{ and } \begin{pmatrix} 5+6i \\3-17i \end{pmatrix}} { }
form a basis.

}
{} {}




\inputexercise
{4}
{

Let $\Q^n$ be the $n$-dimensional standard vector space over $\Q$ and let $v_1 , \ldots , v_n \in \Q^n$ be a family of vectors. Prove that this family is a $\Q$-basis of $\Q^n$ if and only if the same family, considered as a family in $\R^n$, is a $\R$-basis of $\R^n$.

}
{} {}




\inputexercise
{3}
{

Let $K$ be a field and let
\mathdisp {\begin{pmatrix} a_1 \\\vdots\\ a_n \end{pmatrix} \in K^n} { }
be a nonzero vector. Find a linear system of equations in $n$ variables with $n-1$ equations, whose solution space is exactly
\mathdisp {{ \left\{ \lambda \begin{pmatrix} a_1 \\\vdots\\ a_n \end{pmatrix} \mid \lambda \in K \right\} }} { . }

}
{} {}



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