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I'm having trouble representing a certain linear transformation as a matrix. In general, for a linear transformation that takes in a column vector, I know how to find this representation. However, my problem arises in the fact that this transformation instead takes a 2x2 matrix as its arguments.

$T(\begin{bmatrix} a && b \\ c && d \end{bmatrix}) = \begin{bmatrix} d && a \\ b && c \end{bmatrix} $

I attempted to apply $T$ to the standard basis of the 2x2 matrices. However, I ended up with a 4x4 matrix (understandable, as there are 4 matrices in that basis). This matrix I came up with was

$ \begin{bmatrix} 0 && 0 && 0 && 1 \\ 1 && 0 && 0 && 0 \\ 0 && 1 && 0 && 0 \\ 0 && 0 && 1 && 0 \\ \end{bmatrix} $

I know with that (incorrect) 4x4 matrix I'm on the verge of the answer with that. Any help is greatly appreciated!

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Your issue is that you're still thinking of the $2\times 2$ matrices as matrices. If you're going to have them be acted upon by transformations, and you need those transformations to be represented by matrices, you need to represent the elements (some $2\times 2$ matrices) of the base vector space (the space of $2\times 2$ matrices) as column vectors.

This may seem confusing because it's no longer clear what space each matrix or vector is associate with. You could attack this problem by adding some notation to make it clear.

Let the vector space that the $2\times 2$ matrices act upon be denoted $V$. Then let $M^{2\times 2}$ denote the set of $2\times 2$ matrices associated with linear transformations on $V$. Then what you have could be expressed like this:

$$\begin{bmatrix} a & b \\ c & d \end{bmatrix}_V \equiv \begin{bmatrix} a \\ b \\ c \\ d \end{bmatrix}_{M^{2\times 2}}$$

The subscript says what vector space each matrix or column vector is associated with.

Then your answer is merely

$$\begin{bmatrix} 0 & 0 & 0 & 1 \\ 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \end{bmatrix}_{M^{2\times 2}}$$

which is correct.

This approach of using subscripts to denote what vector space the matrix or column vector is associated with is by no means standard. I do think this is one reason why higher-level treatments of linear algebra sometimes don't even use matrices: when you need to talk about a set of matrices itself as a vector space, things get a little confusing.

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  • $\begingroup$ I had the same confusion! I was thinking how is a $4\times 4$ transformation matrix going to left multiply with a $2\times 2$ element of $M_{2\times 2}$. For this the elements have to be represented as the corresponding $4\times 1$ column vectors. Thanks for clarifying. $\endgroup$ – Rhaldryn Jun 4 '18 at 4:46
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Let the basis be $e_1 = E_{11}, e_2 = E_{12}, e_3 = E_{21}, e_4 = E_{22}$.

Then $T e_1 = e_2, T e_2 = e_3, T e_3 = e_4, T e_4 = e_1$.

So, your representation is correct, assuming your basis matches mine.

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