Construct a new solution to Exercise B.T50 along the following outline. From the $n\times n$ matrix $A$, construct the linear transformation $\ltdefn{T}{\complex{n}}{\complex{n}}$, $\lt{T}{\vect{x}}=A\vect{x}$. Use Theorem NI, Theorem IMILT and Theorem ILTIS to translate between the nonsingularity of $A$ and the surjectivity/injectivity of $T$. Then apply Theorem ILTB and Theorem SLTB to connect these properties with bases.