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## Homework Statement

[tex]f: K^{3} \rightarrow K^{4}[/tex] is a linear transformation of vector spaces:

[tex]K^{3} = \left\langle \vec{e}_{1}, \vec{e}_{2}, \vec{e}_{3} \right\rangle[/tex]

and

[tex]K^{4} = \left\langle \vec{e}^{*}_{1}, \vec{e}^{*}_{2}, \vec{e}^{*}_{3}, \vec{e}^{*}_{4} \right \rangle[/tex]

as well as:

[tex] f(\vec{e}_{1}) = \vec{e}^{*}_{1} - \vec{e}^{*}_{2} + \vec{e}^{*}_{3} - \vec{e}^{*}_{4}[/tex],

[tex] f(\vec{e}_{2}) = \vec{e}^{*}_{1} - 2 \vec{e}^{*}_{3}[/tex],

[tex] f(\vec{e}_{1}) = \vec{e}^{*}_{2} - 3 \vec{e}^{*}_{3} + \vec{e}^{*}_{4}[/tex].

Determine a matrix A so that for all [tex] x \in K^{3}[/tex] so that

[tex] f(x) = Ax [/tex]

Determine the kernel and image of f.

## Homework Equations

## The Attempt at a Solution

well I assumed the following:

[tex]K^{3} = \left\langle \vec{e}_{1} \vec{e}_{2} \vec{e}_{3} \right\rangle \[

=

\left[ {\begin{array}{ccc}

1 & 0 & 0 \\

0 & 1 & 0 \\

0 & 0 & 1 \\

\end{array} } \right]

\]

[/tex]

[tex][tex]K^{4} = \left\langle \vec{e}^{*}_{1} \vec{e}^{*}_{2} \vec{e}^{*}_{3} \vec{e}^{*}_{4} \right \rangle \[

=

\left[ {\begin{array}{cccc}

1 & 0 & 0 & 0 \\

0 & 1 & 0 & 0 \\

0 & 0 & 1 & 0 \\

0 & 0 & 0 & 1 \\

\end{array} } \right]

\]

[/tex]

[tex] f(\vec{e}_{1}) \[

=

\left[ {\begin{array}{c}

1 \\

-1 \\

1 \\

-1 \\

\end{array} } \right]

\]

[/tex],

[tex] f(\vec{e}_{2}) \[

=

\left[ {\begin{array}{c}

1 \\

0 \\

-2 \\

0 \\

\end{array} } \right]

\]

[/tex],

[tex] f(\vec{e}_{1}) \[

=

\left[ {\begin{array}{c}

0 \\

1 \\

-3 \\

1 \\

\end{array} } \right]

\]

[/tex].

[tex] f(x) \[

=

\left[ {\begin{array}{ccc}

1 & 1 & 0 \\

-1 & 0 & 1 \\

1 & 2 & -3 \\

-1 & 0 & 1 \\

\end{array} } \right]

\]

\[

= Ax = A

\left[ {\begin{array}{ccc}

1 & 0 & 0 \\

0 & 1 & 0 \\

0 & 0 & 1 \\

\end{array} } \right]

\]

= A = \[

\left[ {\begin{array}{ccc}

1 & 1 & 0 \\

-1 & 0 & 1 \\

1 & 2 & -3 \\

-1 & 0 & 1 \\

\end{array} } \right]

\]

[/tex]

so that's A, but I don't think it can be right for a start its not 4D.

I know how to get the kernel and image but I don't really know how else to start this problem

Last edited: