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Eclair_de_XII
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Homework Statement
"Find ##S_\alpha## where ##S: M_{2×2}(ℝ)→M_{2×2}(ℝ)## is defined by ##S(A)=A^T##.
Homework Equations
##A^T=\begin{pmatrix}
a_{11} & a_{21} \\
a_{12} & a_{22}
\end{pmatrix}##
##\alpha= \{
{\begin{pmatrix}
1 & 0 \\
0 & 0 \end{pmatrix},
\begin{pmatrix}
0 & 0 \\
1 & 0 \end{pmatrix},
\begin{pmatrix}
0 & 1 \\
0 & 0 \end{pmatrix},
\begin{pmatrix}
0 & 0 \\
0 & 1 \end{pmatrix}} \}##
The Attempt at a Solution
I've found ##S_\alpha## before where the vectors can be expressed as columns, but i have no experience with finding the transformation matrices for actual matrices. Can anyone help me? This is as far as I got:
##S(A)=A^T=\begin{pmatrix}
a_{11} & a_{21} \\
a_{12} & a_{22} \end{pmatrix}=a_{11}\begin{pmatrix}1 & 0 \\
0 & 0 \end{pmatrix}+a_{12}\begin{pmatrix}
0 & 0 \\
1 & 0 \end{pmatrix}+a_{21}\begin{pmatrix}
0 & 1 \\
0 & 0 \end{pmatrix}+a_{22}\begin{pmatrix}
0 & 0 \\
0 & 1 \end{pmatrix}
##
How would I form this into a four-dimensional basis?
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