Proof of Theorem FTMR

Let $B=\set{\vectorlist{u}{n}}$ be the basis of $U$. Since $\vect{u}\in U$, there are scalars $\scalarlist{a}{n}$ such that \begin{equation*} \vect{u}=\lincombo{a}{u}{n} \end{equation*}

Then, \begin{align*} &\matrixrep{T}{B}{C}\vectrep{B}{\vect{u}}\\ &= \left[ \left.\vectrep{C}{\lt{T}{\vect{u}_1}}\right| \left.\vectrep{C}{\lt{T}{\vect{u}_2}}\right| \left.\vectrep{C}{\lt{T}{\vect{u}_3}}\right| \ldots \left|\vectrep{C}{\lt{T}{\vect{u}_n}}\right. \right] \vectrep{B}{\vect{u}} &&\knowl{./knowls/definition.MR.knowl}{\text{Definition MR}}\\ &= \left[ \left.\vectrep{C}{\lt{T}{\vect{u}_1}}\right| \left.\vectrep{C}{\lt{T}{\vect{u}_2}}\right| \left.\vectrep{C}{\lt{T}{\vect{u}_3}}\right| \ldots \left|\vectrep{C}{\lt{T}{\vect{u}_n}}\right. \right] \colvector{a_1\\a_2\\a_3\\\vdots\\a_n} &&\knowl{./knowls/definition.VR.knowl}{\text{Definition VR}}\\ &= a_1\vectrep{C}{\lt{T}{\vect{u}_1}}+ a_2\vectrep{C}{\lt{T}{\vect{u}_2}}+ \cdots+ a_n\vectrep{C}{\lt{T}{\vect{u}_n}} &&\knowl{./knowls/definition.MVP.knowl}{\text{Definition MVP}}\\ &= \vectrep{C}{ a_1\lt{T}{\vect{u}_1}+ a_2\lt{T}{\vect{u}_2}+ a_3\lt{T}{\vect{u}_3}+\cdots+ a_n\lt{T}{\vect{u}_n} }&&\knowl{./knowls/theorem.LTLC.knowl}{\text{Theorem LTLC}}\\ &= \vectrep{C}{ \lt{T}{ \lincombo{a}{u}{n} }}&&\knowl{./knowls/theorem.LTLC.knowl}{\text{Theorem LTLC}}\\ &= \vectrep{C}{\lt{T}{\vect{u}}} \end{align*}

The alternative conclusion is obtained as \begin{align*} \lt{T}{\vect{u}} &=\lt{I_V}{\lt{T}{\vect{u}}} &&\knowl{./knowls/definition.IDLT.knowl}{\text{Definition IDLT}}\\ &=\lt{\left(\compose{\vectrepinvname{C}}{\vectrepname{C}}\right)}{\lt{T}{\vect{u}}} &&\knowl{./knowls/definition.IVLT.knowl}{\text{Definition IVLT}}\\ &=\vectrepinv{C}{\vectrep{C}{\lt{T}{\vect{u}}}} &&\knowl{./knowls/definition.LTC.knowl}{\text{Definition LTC}}\\ &=\vectrepinv{C}{\matrixrep{T}{B}{C}\left(\vectrep{B}{\vect{u}}\right)}\\ \end{align*}