Kurs:Mathematik für Anwender (Osnabrück 2011-2012)/Teil I/Arbeitsblatt 11/en/latex

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\zwischenueberschrift{Warm-up-exercises}




\inputexercise
{}
{

Determine explicitly the column rank and the row rank of the matrix
\mathdisp {\begin{pmatrix} 3 & 2 & 6 \\ 4 & 1 & 5 \\6 & -1 & 3 \end{pmatrix}} { . }
Describe linear dependencies \zusatzklammer {if they exist} {} {} between the rows and between the columns of the matrix.

}
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\inputexercise
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{

Show that the elementary operations on the rows do not change the column rank.

}
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\inputexercise
{}
{

Compute the determinant of the matrix
\mathdisp {\begin{pmatrix} 1+3i & 5-i \\ 3-2i & 4+i \end{pmatrix}} { . }

}
{} {}




\inputexercise
{}
{

Compute the determinant of the matrix
\mathdisp {\begin{pmatrix} 1 & 3 & 5 \\ 2 & 1 & 3 \\8 & 7 & 4 \end{pmatrix}} { . }

}
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\inputexercise
{}
{

Prove by induction that the determinant of an upper triangular matrix is equal to the product of the diagonal elements.

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\inputexercise
{}
{

Check the multi-linearity and the property to be alternating, directly for the determinant of a $3\times3$-matrix.

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\inputexercise
{}
{

Let $M$ be the following square matrix
\mathdisp {M= \begin{pmatrix} A & B \\ 0 & D \end{pmatrix}} { , }
where \mathkor {} {A} {and} {D} {} are square matrices. Prove that
\mathl{\det M = \det A \cdot \det D}{.}

}
{} {}




\inputexercise
{}
{

Determine for which $x \in {\mathbb C}$ the matrix
\mathdisp {\begin{pmatrix} x^2+x & -x \\ -x^3+2x^2+5x-1 & x^2-x \end{pmatrix}} { }
is invertible.

}
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\inputexercise
{}
{






\bild{ \begin{center}
\includegraphics[width=5.5cm]{\bildeinlesung {Linalg_parallelogram_area.eps} }
\end{center}
\bildtext {} }

\bildlizenz { Linalg parallelogram area.png } {Nicholas Longo} {Thenub314} {Commons} {CC-by-sa 2.5} {}

Use the image to convince yourself that, given two vectors \mathkor {} {(x_1,y_1)} {and} {(x_2,y_2)} {,} the determinant of the $2\times 2$-matrix defined by these vectors is equal \zusatzklammer {up to sign} {} {} to the area of the plane \stichwort {parallelogram} {} spanned by the vectors.

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\inputexercise
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{

Prove that you can develop the determinant according to each row and each column.

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\inputexercise
{}
{

Let $K$ be a field and $m,n,p \in \N$. Prove that the transpose of a matrix satisfy the following properties \zusatzklammer {where $A,B \in \operatorname{Mat}_{ m \times n } (K)$, $C \in \operatorname{Mat}_{ n \times p } (K)$ and $s \in K$} {.} {.} \aufzaehlungvier{ ${ ({ A^{ \text{tr} } } )^{ \text{tr} } } =A$. }{ ${ (A+B)^{ \text{tr} } } = { A ^{ \text{tr} } } + { B^{ \text{tr} } }$. }{ ${ (s A) ^{ \text{tr} } } = s \cdot { A^{ \text{tr} } }$. }{ ${ (A \circ C)^{ \text{tr} } } = { C^{ \text{tr} } } \circ { A ^{ \text{tr} } }$. }

}
{} {}




\inputexercise
{}
{

Compute the determinant of the matrix
\mathdisp {\begin{pmatrix} 0 & 2 & 7 \\ 1 & 4 & 5 \\6 & 0 & 3 \end{pmatrix}} { , }
by developing the matrix along every column and along every row.

}
{} {}




\inputexercise
{}
{

Compute the determinant of all the $3\times 3$-matrices, such that in each column and in each row there are exactly one $1$ and two $0$s.

}
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\inputexercise
{}
{

Let $z \in {\mathbb C}$ and let \maabbeledisp {} {{\mathbb C}} {{\mathbb C} } {w} {zw } {,} be the associated multiplication. Compute the determinant of this map, considering it as a real-linear map \maabb {} {\R^2} {\R^2 } {.}

}
{} {}

The next exercises require the following definition.

Let $K$ be a field and let $V$ be a $K$-vector space. For
\mathl{a \in K}{} the linear map \maabbeledisp {\varphi} {V} {V } {v} {av } {,} is called the \definitionswort {stretching}{} \zusatzklammer {or \definitionswort {homothety}{}} {} {} with \stichwort {extension factor} {} $a$.





\inputexercise
{}
{

What is the determinant of a homothety?

}
{} {}




\inputexercise
{}
{

Check the multiplication theorem for determinants of two homotheties on a finite-dimensional vector space.

}
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\inputexercise
{}
{

Check the multiplication theorem for determinants of the following matrices
\mathdisp {A = \begin{pmatrix} 5 & 7 \\ 2 & -4 \end{pmatrix} \text{ and } B = \begin{pmatrix} -3 & 1 \\ 6 & 5 \end{pmatrix}} { . }

}
{} {}






\zwischenueberschrift{Hand-in-exercises}




\inputexercise
{4}
{

Let $K$ be a field and let \mathkor {} {V} {and} {W} {} be vector spaces over $K$ of dimensions \mathkor {} {n} {and} {m} {.} Let \maabbdisp {\varphi} {V} {W } {} be a linear map, described by the matrix
\mathl{M \in \operatorname{Mat}_{ m \times n } (K)}{} with respect to two bases. Prove that
\mathdisp {\operatorname{rang} \, \varphi = \operatorname{rang} \, M} { . }

}
{} {}




\inputexercise
{3}
{

Compute the determinant of the matrix
\mathdisp {\begin{pmatrix} 1+i & 3-2i & 5 \\ i & 1 & 3-i \\2i & -4-i & 2+i \end{pmatrix}} { . }

}
{} {}




\inputexercise
{4}
{

Compute the determinant of the matrix
\mathdisp {A=\begin{pmatrix} 2 & 1 & 0 & -2 \\ 1 & 3 & 3 & -1 \\ 3 & 2 & 4 & -3 \\ 2 & -2 & 2 & 3 \end{pmatrix}\,.} { }

}
{} {}




\inputexercise
{2}
{

Compute the determinant of the elementary matrices.

}
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\inputexercise
{5}
{

Check the multiplication theorem for determinants of the following matrices
\mathdisp {A = \begin{pmatrix} 3 & 4 & 7 \\ 2 & 0 & -1 \\1 & 3 & 4 \end{pmatrix} \text{ and } B = \begin{pmatrix} -2 & 1 & 0 \\ 2 & 3 & 5 \\2 & 0 & -3 \end{pmatrix}} { . }

}
{} {}



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