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Suppose i have a vector space $V=\mathbb{F}^{3\times2}$ of $3\times2$ matrices and a vector space $W=\mathbb{F}^{2\times2}$ of $2\times2$ matrices over the field $\mathbb{F}$.

Any Linear transformation $T: V\to W$ would take a $3\times2$ matrix and transform it into a $2\times2$.

I tried to compare this with the fact that multiplying a $2\times3$ matrix by a $3\times2$ matrix also results in a $2\times2$ matrix.

But obviously this $2\times3$ matrix is not the matrix representation of $T$.

To better understand, i defined a standard basis for $V$ and made some algebra resulting in this fact :

Where the elements of the $6\times1$ matrix are the vectors from the standard basis of $\mathbb{F}^{3\times2}$ after being transformed by $T$.

By all of this, i found that there are at least two ways to implement matrix multiplication to result in a $2\times2$ matrix.

My question :

Is the ( $1\times6$ times $6\times1$ ) matrix multiplication the only one that is equivalent to the ($2\times3$ times $3\times2$) matrix multiplication ?

I can't find detailed information about matrix multplication, even on linear algebra books ( even wikipedia doesn't mention the equivalent matrix multiplications i found out empirically ).

Is there any topic name, site, resource, book that can teach much more than the algorithm approach (row dot column ) of matrix multiplication ?

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I didn't understand your explicit question, but here's a comment about $T$ itself: $T$ is a linear transformation from a six-dimensional vector space $V$ to a four-dimensional vector space $W$. Therefore if you choose a basis for $V$ and a basis for $W$, the corresponding matrix representation of $T$ will be a $4\times6$ matrix. – Greg Martin Apr 23 '13 at 7:45

1 Answer 1

Not all linear maps between matrix spaces arise from one-sided matrix multiplication.

Indeed, the space of linear maps from $F^{3\times 2}$ to $F^{2\times 2}$ is $24$-dimensional ($6\cdot 4=24$). Some of them are realized as matrix multiplication $M\mapsto AM$ where $A\in F^{2\times 3}$. However, the matrix $A$ has only $6$ entires, so the linear spaces of all such transformations is $6$-dimensional: it is a proper subspace of $L(F^{3\times 2},F^{2\times 2})$.

However, every linear map from $F^{3\times 2}$ to $F^{2\times 2}$ can be written as a finite sum of two-sided multiplication maps $M\mapsto AMB$. Indeed, $24$ terms suffice: $$M\mapsto \sum_{j=1}^{24} A_j MB_j$$ where each matrix $A_j$ and $B_j$ has one nonzero entry. To see why, observe that by placing $1$ at one entry of $A$ and one entry of $B$ one can move any given entry of $M$ into any desired position in $AMB$.

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