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I know that in order to find transformation matrix with respect to a basis, I need to apply the transformation to said basis and the result is the column of the transformation matrix.

But what happens when the linear transformation is applied to matrices and not vectors? then the map is also a matrix. How am i suppose to write the matrix as a column?

more specifically, I need to find the transformation matrix of $T(\begin{pmatrix} a & b \\ c & d \end{pmatrix}) = \begin{pmatrix} 2ia & b+ci \\ c+bi & 2id \end{pmatrix}$

$T: M_2(\mathbb C) -> M_2(\mathbb C)$.

so with respect to standard basis:

$T(\begin{pmatrix} 1 & 0 \\ 0 & 0 \end{pmatrix}) = \begin{pmatrix} 2i & 0 \\ 0 & 0 \end{pmatrix}$ what does this mean? Is the first column of the transformation matrix $\begin{pmatrix} 2i & 0 \\ 0 & 0 \end{pmatrix}$? That makes no sense.

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That' correct. Your transformation matrix is here a tensor –  aflous Dec 22 '13 at 18:01

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up vote 2 down vote accepted

The matrix of a transformation is the matrix that turns the vector of coordinates of the input into the vector of coordinates of the output in certain bases. Coordinate vectors are always column vectors of some $\mathbb{R}^n$.

For example: In the standard basis $\{\begin{bmatrix}1&0\\0&0\end{bmatrix},\begin{bmatrix}0&1\\0&0\end{bmatrix},\begin{bmatrix}0&0\\1&0\end{bmatrix},\begin{bmatrix}0&0\\0&1\end{bmatrix}\}$, of the space of matrices. The coordinate vector of the matrix $\begin{bmatrix}a&b\\c&d\end{bmatrix}$, is the column $\begin{bmatrix}a\\b\\c\\d\end{bmatrix}$.

In your case the transformation is $$T(\begin{bmatrix}a&b\\c&d\end{bmatrix})=\begin{bmatrix}2ia&b+ci\\c+bi&2id\end{bmatrix}$$

When you compute the matrix of this transformation in the standard basis you get $$A_T:=\begin{bmatrix}2i&0&0&0\\0&1&i&0\\0&i&1&0\\0&0&0&2i\end{bmatrix}$$

What this means is that when you multiply the vector of coordinates $$\begin{bmatrix}a\\b\\c\\d\end{bmatrix}$$ of some vector $\begin{bmatrix}a&b\\c&d\end{bmatrix}$ of your space, by $A_T$ you get the vector of coordinates $$\begin{bmatrix}2ia\\b+ci\\c+bi\\2di\end{bmatrix}=A_t\begin{bmatrix}a\\b\\c\\d\end{bmatrix}$$

of the output of $T$. This is, the vector of coordinates of the vector $\begin{bmatrix}2ai&b+ci\\c+bi&2di\end{bmatrix}$ in the standard basis.

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