Proof of Theorem EMMVP

We are assuming $A\vect{x}=B\vect{x}$ for all $\vect{x}\in\complex{n}$, so we can employ this equality for any choice of the vector $\vect{x}$. However, we will limit our use of this equality to the standard unit vectors, $\vect{e}_j$, $1\leq j\leq n$ (Definition SUV). For all $1\leq j\leq n$, $1\leq i\leq m$, \begin{align*} &\matrixentry{A}{ij}\\ &= 0\matrixentry{A}{i1}+ \cdots+ 0\matrixentry{A}{i,j-1}+ 1\matrixentry{A}{ij}+ 0\matrixentry{A}{i,j+1}+ \cdots+ 0\matrixentry{A}{in} &&\knowl{./knowls/theorem.PCNA.knowl}{\text{Theorem PCNA}}\\ &= \vectorentry{\vect{e}_j}{1}\matrixentry{A}{i1}+ \vectorentry{\vect{e}_j}{2}\matrixentry{A}{i2}+ \vectorentry{\vect{e}_j}{3}\matrixentry{A}{i3}+ \cdots+ \vectorentry{\vect{e}_j}{n}\matrixentry{A}{in} &&\knowl{./knowls/property.CMCN.knowl}{\text{Property CMCN}}\\ &= \matrixentry{A}{i1}\vectorentry{\vect{e}_j}{1}+ \matrixentry{A}{i2}\vectorentry{\vect{e}_j}{2}+ \matrixentry{A}{i3}\vectorentry{\vect{e}_j}{3}+ \cdots+ \matrixentry{A}{in}\vectorentry{\vect{e}_j}{n} &&\knowl{./knowls/definition.SUV.knowl}{\text{Definition SUV}}\\ &= \vectorentry{A\vect{e}_j}{i}&&\knowl{./knowls/definition.MVP.knowl}{\text{Definition MVP}}\\ &= \vectorentry{B\vect{e}_j}{i}&&\text{Hypothesis}\\ &= \matrixentry{B}{i1}\vectorentry{\vect{e}_j}{1}+ \matrixentry{B}{i2}\vectorentry{\vect{e}_j}{2}+ \matrixentry{B}{i3}\vectorentry{\vect{e}_j}{3}+ \cdots+ \matrixentry{B}{in}\vectorentry{\vect{e}_j}{n}&&\knowl{./knowls/definition.MVP.knowl}{\text{Definition MVP}}\\ &= \vectorentry{\vect{e}_j}{1}\matrixentry{B}{i1}+ \vectorentry{\vect{e}_j}{2}\matrixentry{B}{i2}+ \vectorentry{\vect{e}_j}{3}\matrixentry{B}{i3}+ \cdots+ \vectorentry{\vect{e}_j}{n}\matrixentry{B}{in}&&\knowl{./knowls/property.CMCN.knowl}{\text{Property CMCN}}\\ &= 0\matrixentry{B}{i1}+ \cdots+ 0\matrixentry{B}{i,j-1}+ 1\matrixentry{B}{ij}+ 0\matrixentry{B}{i,j+1}+ \cdots+ 0\matrixentry{B}{in} &&\knowl{./knowls/definition.SUV.knowl}{\text{Definition SUV}}\\ &=\matrixentry{B}{ij} &&\knowl{./knowls/theorem.PCNA.knowl}{\text{Theorem PCNA}} \end{align*}

So by Definition ME the matrices $A$ and $B$ are equal, as desired.