An example asks me to define $T: M_{2\times2}(F) \to M_{2\times2}$ by $T(A) = A^t$. Compute $[T]_\alpha$
$\alpha$ is the standard ordered basis of $2\times 2$ matrices.
To find the transformation I performed the transformation on all four elements of $\alpha$, for example $T(\begin{pmatrix} 1&0\\0&0\\\end{pmatrix}) = (\begin{pmatrix} 1&0\\0&0\\\end{pmatrix}) $ and so on. Basically the matrices are all the same before and after the transformation. Accordingly, the textbook's answer is that you arrange all the results $A^t$ into a $4\times4$ matrix:
$\begin{pmatrix} 1&0&0&0\\0&0&1&0\\0&1&0&0\\0&0&0&1\end{pmatrix}$
I don't understand this what means, and I've been trying to. How can this be the transformation $T$ on $\alpha$ if we're not even able to multiply the basis by it? What I mean is, we can't multiply this transformation matrix by anything in $\alpha$ since one is a $4\times4$ and one is a $2\times 2$.
Also, why doesn't this mean that any arbitrary $2\times 2$ matrix can be raised to the power of $t$ by simply multiplying by this matrix? (As per the previous paragraph, it can't, but it should).