Solution MR.T20 Robert Beezer

Given the nonsingular $n\times n$ matrix $A$, create the linear transformation $\ltdefn{T}{\complex{n}}{\complex{n}}$ defined by $\lt{T}{\vect{x}}=A\vect{x}$. Then \begin{align*} A\text{ nonsingular} &\iff A\text{ invertible} &&\knowl{./knowls/theorem.NI.knowl}{\text{Theorem NI}}\\ &\iff T\text{ invertible} &&\knowl{./knowls/theorem.IMILT.knowl}{\text{Theorem IMILT}}\\ &\iff T\text{ injective and surjective} &&\knowl{./knowls/theorem.ILTIS.knowl}{\text{Theorem ILTIS}}\\ &\iff C\text{ linearly independent, and} &&\knowl{./knowls/theorem.ILTB.knowl}{\text{Theorem ILTB}}\\ &\quad\quad\quad\quad C\text{ spans }\complex{n} &&\knowl{./knowls/theorem.SLTB.knowl}{\text{Theorem SLTB}}\\ &\iff C\text{ basis for }\complex{n} &&\knowl{./knowls/definition.B.knowl}{\text{Definition B}} \end{align*}