 Mohamed Mooney

2022-06-24

For a transformation
$F\left(X\right)={X}^{T}$
on $V={M}_{3,3}\left(\mathbb{R}\right)$, apparently the matrix of $F$ is a $9×9$ matrix. How can this be possible? Isn't the definition that
$F\left(X\right)=AX?$
, so if $A$ is the $9×9$ matrix, we can't multiply a $9×9$ matrix with a $3×3$, can we? assumintdz

For any linear transformation $T:V\to W$ over finite dimensional real vector spaces if we consider fixed ordered bases $B,{B}^{\prime }$ of $V$ and $W$ respectively, then we can find the coordinate vectors/expansion in terms of these bases.
This directly follows from the definition of the bases. Given $X\in V$ and $B=\left\{{v}_{1},\dots ,{v}_{n}\right\}$ clearly $X=\sum {\alpha }_{i}{v}_{i}$ is a unique way to express $X$ in terms of the ${v}_{i}$. If this way was not unique then linear independence of $B$ will be contradicted. So we may associate $\left[X\right]={\left(\begin{array}{ccc}{\alpha }_{1}& \cdots & {\alpha }_{n}\end{array}\right)}^{T}$ with $X$ in a one to one way. Likewise coordinates can be associated with every vector in $V$. This association essentially allows us to identify an element of the vector space $V$ with the vector space ${\mathbb{R}}^{\mathrm{dim}V}$. The same can be done in $W$ too with respect to ${B}^{\prime }$.
Now if $A$ is the matrix of $T$ with respect to these bases, then a very beautiful result (which is not difficult to prove also) says that as $X\to T\left(X\right)$, the corresponding coordinates of $X$, say $\left[X\right]={\left(\begin{array}{ccc}{\alpha }_{1}& \cdots & {\alpha }_{n}\end{array}\right)}^{T}$, go to the corresponding coordinates of $T\left(X\right)$, which are precisely given by $A{\left(\begin{array}{ccc}{\alpha }_{1}& \cdots & {\alpha }_{n}\end{array}\right)}^{T}$. Essentially, the role of the action of $T$ is played by multiplication by $A$ in the coordinate world. So the correct interpretation is not that $X\to AX$ but $\left[X\right]\to A\left[X\right]$. It is not that the matrix and the transformation are identical, but under the identification of vectors by their coordinates, the behavior of the transformation matches the action of multiplication by the matrix. The same holds if we are working over any arbitrary field instead of $\mathbb{R}$.
As an illustration in your case let $V=W={M}_{3,3}\left(\mathbb{R}\right)$ and $B={B}^{\prime }=\left\{{e}_{11},{e}_{12},{e}_{13},{e}_{21},{e}_{22},{e}_{23},{e}_{31},{e}_{32},{e}_{33}\right\}$ be the fixed ordered basis of $V$ where ${e}_{ij}$ is the $3×3$ matrix which has $1$ at the $\left(i,j\right)th$ place and $0$'s elsewhere.
It is now clear that the matrix of $F$ with respect to this basis is
$A=\left(\begin{array}{ccccccccc}1& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 1& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 1& 0& 0\\ 0& 1& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 1& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 1& 0\\ 0& 0& 1& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 1& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 1\end{array}\right)$
If we have some $3×3$ matrix $X=\left(\begin{array}{ccc}1& 2& 3\\ 4& 5& 6\\ 7& 8& 9\end{array}\right)$ then its coordinates will be
${\left(\begin{array}{ccccccccc}1& 2& 3& 4& 5& 6& 7& 8& 9\end{array}\right)}^{T}$
Now since $X=\left(\begin{array}{ccc}1& 2& 3\\ 4& 5& 6\\ 7& 8& 9\end{array}\right)\to F\left(X\right)=\left(\begin{array}{ccc}1& 4& 7\\ 2& 5& 8\\ 3& 6& 9\end{array}\right)$, under $F$, so it must happen that their coordinates also change in the same fashion under $A$.
So we must have
$\left[X\right]={\left(\begin{array}{ccccccccc}1& 2& 3& 4& 5& 6& 7& 8& 9\end{array}\right)}^{T}\to A\left[X\right]={\left(\begin{array}{ccccccccc}1& 4& 7& 2& 5& 8& 3& 6& 9\end{array}\right)}^{T}$
which can be verified by direct multiplication.
Also, in the same vein, if you compose linear transformations ${T}_{1},{T}_{2}$ then their effect is the same as multiplying the corresponding matrices. This is the real reason behind defining matrix multiplication in the fashion that we do so.

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