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Homework Statement
f: K^{3} \rightarrow K^{4} is a linear transformation of vector spaces:
K^{3} = \left\langle \vec{e}_{1}, \vec{e}_{2}, \vec{e}_{3} \right\rangle
and
K^{4} = \left\langle \vec{e}^{*}_{1}, \vec{e}^{*}_{2}, \vec{e}^{*}_{3}, \vec{e}^{*}_{4} \right \rangle
as well as:
f(\vec{e}_{1}) = \vec{e}^{*}_{1} - \vec{e}^{*}_{2} + \vec{e}^{*}_{3} - \vec{e}^{*}_{4},
f(\vec{e}_{2}) = \vec{e}^{*}_{1} - 2 \vec{e}^{*}_{3},
f(\vec{e}_{1}) = \vec{e}^{*}_{2} - 3 \vec{e}^{*}_{3} + \vec{e}^{*}_{4}.
Determine a matrix A so that for all x \in K^{3} so that
f(x) = Ax
Determine the kernel and image of f.
Homework Equations
The Attempt at a Solution
well I assumed the following:
K^{3} = \left\langle \vec{e}_{1} \vec{e}_{2} \vec{e}_{3} \right\rangle \[<br /> =<br /> \left[ {\begin{array}{ccc}<br /> 1 & 0 & 0 \\<br /> 0 & 1 & 0 \\<br /> 0 & 0 & 1 \\<br /> \end{array} } \right]<br /> \]<br />
K^{4} = \left\langle \vec{e}^{*}_{1} \vec{e}^{*}_{2} \vec{e}^{*}_{3} \vec{e}^{*}_{4} \right \rangle \[<br /> =<br /> \left[ {\begin{array}{cccc}<br /> 1 &amp; 0 &amp; 0 &amp; 0 \\<br /> 0 &amp; 1 &amp; 0 &amp; 0 \\<br /> 0 &amp; 0 &amp; 1 &amp; 0 \\<br /> 0 &amp; 0 &amp; 0 &amp; 1 \\<br /> \end{array} } \right]<br /> \]<br /><br /> f(\vec{e}_{1}) \[<br /> =<br /> \left[ {\begin{array}{c}<br /> 1 \\<br /> -1 \\<br /> 1 \\<br /> -1 \\<br /> \end{array} } \right]<br /> \]<br />,<br /> <br /> f(\vec{e}_{2}) \[<br /> =<br /> \left[ {\begin{array}{c}<br /> 1 \\<br /> 0 \\<br /> -2 \\<br /> 0 \\<br /> \end{array} } \right]<br /> \]<br />,<br /> <br /> f(\vec{e}_{1}) \[<br /> =<br /> \left[ {\begin{array}{c}<br /> 0 \\<br /> 1 \\<br /> -3 \\<br /> 1 \\<br /> \end{array} } \right]<br /> \]<br />.<br /> <br /> f(x) \[<br /> =<br /> \left[ {\begin{array}{ccc}<br /> 1 &amp; 1 &amp; 0 \\<br /> -1 &amp; 0 &amp; 1 \\<br /> 1 &amp; 2 &amp; -3 \\<br /> -1 &amp; 0 &amp; 1 \\<br /> \end{array} } \right]<br /> \]<br /> \[<br /> = Ax = A<br /> \left[ {\begin{array}{ccc}<br /> 1 &amp; 0 &amp; 0 \\<br /> 0 &amp; 1 &amp; 0 \\<br /> 0 &amp; 0 &amp; 1 \\<br /> \end{array} } \right]<br /> \]<br /> = A = \[<br /> \left[ {\begin{array}{ccc}<br /> 1 &amp; 1 &amp; 0 \\<br /> -1 &amp; 0 &amp; 1 \\<br /> 1 &amp; 2 &amp; -3 \\<br /> -1 &amp; 0 &amp; 1 \\<br /> \end{array} } \right]<br /> \]<br /><br /> <br /> so that's A, but I don't think it can be right for a start its not 4D.<br /> I know how to get the kernel and image but I don't really know how else to start this problem
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