Proof of Theorem EER

(⇒) Assume that $\vect{v}\in V$ is an eigenvector of $T$ for the eigenvalue $\lambda$. Then \begin{align*} \matrixrep{T}{B}{B}\vectrep{B}{\vect{v}} &=\vectrep{B}{\lt{T}{\vect{v}}}&&\knowl{./knowls/theorem.FTMR.knowl}{\text{Theorem FTMR}}\\ &=\vectrep{B}{\lambda\vect{v}}&&\knowl{./knowls/definition.EELT.knowl}{\text{Definition EELT}}\\ &=\lambda\vectrep{B}{\vect{v}}&&\knowl{./knowls/theorem.VRLT.knowl}{\text{Theorem VRLT}} \end{align*} which by Definition EEM says that $\vectrep{B}{\vect{v}}$ is an eigenvector of the matrix $\matrixrep{T}{B}{B}$ for the eigenvalue $\lambda$.

(⇐) Assume that $\vectrep{B}{\vect{v}}$ is an eigenvector of $\matrixrep{T}{B}{B}$ for the eigenvalue $\lambda$. Then \begin{align*} \lt{T}{\vect{v}} &=\vectrepinv{B}{\vectrep{B}{\lt{T}{\vect{v}}}}&&\knowl{./knowls/definition.IVLT.knowl}{\text{Definition IVLT}}\\ &=\vectrepinv{B}{\matrixrep{T}{B}{B}\vectrep{B}{\vect{v}}}&&\knowl{./knowls/theorem.FTMR.knowl}{\text{Theorem FTMR}}\\ &=\vectrepinv{B}{\lambda\vectrep{B}{\vect{v}}}&&\knowl{./knowls/definition.EEM.knowl}{\text{Definition EEM}}\\ &=\lambda\vectrepinv{B}{\vectrep{B}{\vect{v}}}&&\knowl{./knowls/theorem.ILTLT.knowl}{\text{Theorem ILTLT}}\\ &=\lambda\vect{v}&&\knowl{./knowls/definition.IVLT.knowl}{\text{Definition IVLT}} \end{align*} which by Definition EELT says $\vect{v}$ is an eigenvector of $T$ for the eigenvalue $\lambda$.