Let $\displaystyle T$ be a linear operator one the finite-dimensional space $\displaystyle V$. Suppose there is a linear operator $\displaystyle U $on $\displaystyle V$ such that $\displaystyle TU$ =$\displaystyle I$. Prove that $\displaystyle T$ is invertible and $\displaystyle U =T^-1$ Give an example which shows that this is false when $\displaystyle V$ is not finite-dimensional.

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