I am currently working with using augmented matrices to solve for systems of equations. This one question came up and I am confused why its true :
Suppose that $A$ and $B$ are matrices having $n$ rows. Prove that $M(A|B)=(MA|MB)$ for any $m \times n$ matrix $M$
Does anyone have an explanation for why this is true?
If $A$ and $B$ are matrices over a field $F$ with the $n$ rows then we can think of $A:F^k\to F^n$ and $B:F^l\to F^n$ as linear maps. Here $k$ and $l$ are the number of columns of $A$ and $B$, respectively. Since these linear maps have the same target space we can build a new linear map $A|B:F^k\times F^l\to F^n$ using the formula $A|B(v,w)=Av+Bw$. There is a standard identification between $F^k\times F^l$ and $F^{k+l}$ and the augmented matrix $(A|B)$ is just the matrix for the linear map $A|B$ that is described above.
If $M$ is an $m\times n$ matrix then this can be regarded as a linear map from $F^n\to F^m$. Composing these we get $M(A|B):F^{k}\times F^l\to F^m$. But by examining the formula we find that
$$M(A|B)(v,w)=M(Av+Bw)=MA(v)+MB(w)=MA|MB(v,w)$$
What this means is that the linear maps $M(A|B)$ and $MA|MB$ are equal. Hence the augmented matrices $M(A|B)$ and $(MA|MB)$ are also equal.