Proof of Theorem TMSM

The statement to be proved is an equality of matrices, so we work entry-by-entry and use Definition ME. Notice that the desired equality is of $n\times m$ matrices, and think carefully about the objects involved here, plus the many uses of juxtaposition. For $1\leq i\leq m$, $1\leq j\leq n$, \begin{align*} \matrixentry{\transpose{(\alpha A)}}{ji}&= \matrixentry{\alpha A}{ij}&&\knowl{./knowls/definition.TM.knowl}{\text{Definition TM}}\\ &=\alpha\matrixentry{A}{ij}&&\knowl{./knowls/definition.MSM.knowl}{\text{Definition MSM}}\\ &=\alpha\matrixentry{\transpose{A}}{ji}&&\knowl{./knowls/definition.TM.knowl}{\text{Definition TM}}\\ &=\matrixentry{\alpha\transpose{A}}{ji}&&\knowl{./knowls/definition.MSM.knowl}{\text{Definition MSM}} \end{align*}

Since the matrices $\transpose{(\alpha A)}$ and $\alpha\transpose{A}$ agree at each entry, Definition ME tells us the two matrices are equal.