Proof of Theorem VSPM

While some of these properties seem very obvious, they all require proof. However, the proofs are not very interesting, and border on tedious. We will prove one version of distributivity very carefully, and you can test your proof-building skills on some of the others. We will give our new notation for matrix entries a workout here. Compare the style of the proofs here with those given for vectors in Theorem VSPCV — while the objects here are more complicated, our notation makes the proofs cleaner.

To prove Property DSAM, $(\alpha+\beta)A=\alpha A+\beta A$, we need to establish the equality of two matrices (see Proof Technique GS). Definition ME says we need to establish the equality of their entries, one-by-one. How do we do this, when we do not even know how many entries the two matrices might have? This is where the notation for matrix entries, given in Definition M, comes into play. Ready? Here we go.

For any $i$ and $j$, $1\leq i\leq m$, $1\leq j\leq n$, \begin{align*} \matrixentry{(\alpha+\beta)A}{ij}&= (\alpha+\beta)\matrixentry{A}{ij}&&\knowl{./knowls/definition.MSM.knowl}{\text{Definition MSM}}\\ &=\alpha\matrixentry{A}{ij}+\beta\matrixentry{A}{ij}&&\text{Distributivity in $\complex{\null}$}\\ &=\matrixentry{\alpha A}{ij}+\matrixentry{\beta A}{ij}&&\knowl{./knowls/definition.MSM.knowl}{\text{Definition MSM}}\\ &=\matrixentry{\alpha A+\beta A}{ij}&&\knowl{./knowls/definition.MA.knowl}{\text{Definition MA}} \end{align*}

There are several things to notice here. (1) Each equals sign is an equality of scalars (numbers). (2) The two ends of the equation, being true for any $i$ and $j$, allow us to conclude the equality of the matrices by Definition ME. (3) There are several plus signs, and several instances of juxtaposition. Identify each one, and state exactly what operation is being represented by each.