given vector space $V$ so that $\dim(V)$ is finite , $B$ marked as a base to $V$, and in addition, given linear-transformation $T:V \to V$.
if $\text{rank}([T]_B)=\dim(V)$ how can I prove that $[T]_B$ is a reversible matrix?
2026-02-22 21:19:14.1771795154
How can I prove that $[T]_B$ is a reversible matrix?
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$\text{rank}([T]_{B})=\dim\text{Im} T$. Now $\dim\text{Im}T+\dim\ker T=\dim V$, so $\ker T=\{0\}$. Now $[T^{-1}]_{B}$ is the inverse matrix of $[T]_{B}$.