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Let $T\colon C\to C$ be a linear transformation such that $$T\begin{pmatrix}x\\y\end{pmatrix}= \begin{pmatrix}a&b\\c&d\end{pmatrix}\begin{pmatrix}x\\y\end{pmatrix}$$ Define $$S(x+iy)= (ax+by)+i(cx+dy)$$ Then is $S$ a linear transformation for all $a, b, c, d\in C$? It is a L.T. if $a = d$ and $b=-c$, but doubtful for arbitrary $a, b, c, d$. Please give your precious suggestions. Thank you.

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    Please fix the peculiar thing after "such that" -- I can't read it.2012-12-26
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    After asking over ten questions, please learn how to format your question using LaTeX/MathJax markup.2012-12-26

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