Proof of Theorem MLTCV

The conclusion says a certain matrix exists. What better way to prove something exists than to actually build it? So our proof will be constructive (Proof Technique C), and the procedure that we will use abstractly in the proof can be used concretely in specific examples.

Let $\vectorlist{e}{n}$ be the columns of the identity matrix of size $n$, $I_n$ (Definition SUV). Evaluate the linear transformation $T$ with each of these standard unit vectors as an input, and record the result. In other words, define $n$ vectors in $\complex{m}$, $\vect{A}_i$, $1\leq i\leq n$ by \begin{equation*} \vect{A}_i=\lt{T}{\vect{e}_i} \end{equation*}

Then package up these vectors as the columns of a matrix \begin{equation*} A=\matrixcolumns{A}{n} \end{equation*}

Does $A$ have the desired properties? First, $A$ is clearly an $m\times n$ matrix. Then \begin{align*} \lt{T}{\vect{x}} &=\lt{T}{I_n\vect{x}} &&\knowl{./knowls/theorem.MMIM.knowl}{\text{Theorem MMIM}}\\ &=\lt{T}{\matrixcolumns{e}{n}\vect{x}} &&\knowl{./knowls/definition.SUV.knowl}{\text{Definition SUV}}\\ &=\lt{T}{ \vectorentry{\vect{x}}{1}\vect{e}_1+ \vectorentry{\vect{x}}{2}\vect{e}_2+ \vectorentry{\vect{x}}{3}\vect{e}_3+ \cdots+ \vectorentry{\vect{x}}{n}\vect{e}_n } &&\knowl{./knowls/definition.MVP.knowl}{\text{Definition MVP}}\\ &= \lt{T}{\vectorentry{\vect{x}}{1}\vect{e}_1}+ \lt{T}{\vectorentry{\vect{x}}{2}\vect{e}_2}+ \lt{T}{\vectorentry{\vect{x}}{3}\vect{e}_3}+ \cdots+ \lt{T}{\vectorentry{\vect{x}}{n}\vect{e}_n} &&\knowl{./knowls/definition.LT.knowl}{\text{Definition LT}}\\ &= \vectorentry{\vect{x}}{1}\lt{T}{\vect{e}_1}+ \vectorentry{\vect{x}}{2}\lt{T}{\vect{e}_2}+ \vectorentry{\vect{x}}{3}\lt{T}{\vect{e}_3}+ \cdots+ \vectorentry{\vect{x}}{n}\lt{T}{\vect{e}_n} &&\knowl{./knowls/definition.LT.knowl}{\text{Definition LT}}\\ &= \vectorentry{\vect{x}}{1}{\vect{A}_1}+ \vectorentry{\vect{x}}{2}{\vect{A}_2}+ \vectorentry{\vect{x}}{3}{\vect{A}_3}+ \cdots+ \vectorentry{\vect{x}}{n}{\vect{A}_n} &&\text{Definition of $\vect{A}_i$}\\ &=A\vect{x} &&\knowl{./knowls/definition.MVP.knowl}{\text{Definition MVP}} \end{align*} as desired.