Proof of Theorem TMA

The statement to be proved is an equality of matrices, so we work entry-by-entry and use Definition ME. Think carefully about the objects involved here, and the many uses of the plus sign. Realize too that while $A$ and $B$ are $m\times n$ matrices, the conclusion is a statement about the equality of two $n\times m$ matrices. So we begin with: for $1\leq i\leq n$, $1\leq j\leq m$, \begin{align*} \matrixentry{\transpose{(A+B)}}{ij} &=\matrixentry{A+B}{ji}&&\knowl{./knowls/definition.TM.knowl}{\text{Definition TM}}\\ &=\matrixentry{A}{ji}+\matrixentry{B}{ji}&&\knowl{./knowls/definition.MA.knowl}{\text{Definition MA}}\\ &=\matrixentry{\transpose{A}}{ij}+\matrixentry{\transpose{B}}{ij}&&\knowl{./knowls/definition.TM.knowl}{\text{Definition TM}}\\ &=\matrixentry{\transpose{A}+\transpose{B}}{ij}&&\knowl{./knowls/definition.MA.knowl}{\text{Definition MA}} \end{align*}

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