Exercise I.C.4

Question

Let $V$ be a vector space, and let $\phi \in \mathrm{Hom}({V}	o{V})$. Show that
$$\phi 	ext{ is injective } \iff \phi 	ext{ is surjective } \iff \phi 	ext{ is an isomorphism}$$

Toğrul Topal · Accepted Answer

We use the ring strategy to demonstrate the equivalence.
    \begin{itemize}
    \item[$\implies)$] Suppose that $\phi$ is injective. $\dim(\ker \phi) = 0$. Then $\dim(V) = \dim(\phi(V))$, and so from $\phi(V)\subseteq V$ follows $V=\phi(V)$ and $\phi$ is surjective.
    \item[$\implies)$] Follows by Corollary 1.
    \item[$\implies)$] Follows by definition.
    \end{itemize}

Exercise I.C.4

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Exercise I.C.4

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