Distributive law for matrices

Question

Unless I'm mistaken, matrices follow a distributive law, provided that the dimensions line up, so $X(Y+Z) = XY + XZ$. I'm struggling somewhat to prove this fact, though. Is it enough to say that because matrices represent linear transformations, we're simply applying the linearity of $X$? Or is a more rigorous derivation, likely requiring the summation representation of a matrix, required?

A hint on this would be very helpful. I'm mainly interested on the 'how' here rather than the proof itself, as it's very possible I'm misunderstanding the definition of linearity.

Answer 1

The fact that matrices represent linear functions means that for a matrix $A$ and vectors $u, v$, we have $A(u +v) = A u + A v$. This doesn't tell us anything about what happens when we multiply matrices together.

So yes, to prove distributivity you will need something more. If you consider the summation representation the proof is quite straightforward though.

Answer 2

If you've established:

• the correspondence between matrices and linear transformations
• matrix multiplication corresponds to composition of transformations
• matrix addition corresponds to addition of transformations
• linear transformations satisfy this law

then that is enough to conclude it for matrices as well. Explicitly, if $X$ is a matrix transformation, let $\overline{X}$ denote the associated linear transformation, and you have

$$ \overline{X(Y+Z)} = \overline{X} \circ \overline{(Y+Z)} = \overline{X} \circ \left( \overline{Y} + \overline{Z} \right) = \overline{X} \circ \overline{Y} + \overline{X} \circ \overline{Z} = \overline{XY} + \overline{XZ} = \overline{XY + XZ}$$

Since the associated transformations are equal, the matrices $X(Y+Z)$ and $XY + XZ$ are equal. Observe how all of the bullet points above are used in this calculation.