Exercise 1206Linear algebra › Vector spaces and subspacesL1Added 3 Nov 2026

Isomorphism criterion

#isomorphism#dimension#injective linear map

Statement

Let $E$ and $F$ be finite-dimensional vector spaces with $\dim E=\dim F=n$ , and let $f:E\to F$ be linear. Show that $f$ is an isomorphism if and only if $\ker f=\{0\}$ , if and only if $\operatorname{Im} f=F$ .

Hint

Use the rank-nullity theorem: $\dim\ker f+\dim\operatorname{Im} f=n$ .

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Solution

Injectivity. $\ker f=\{0\}$ iff $\dim\ker f=0$ iff (by rank-nullity) $\dim\operatorname{Im} f=n=\dim F$ iff $\operatorname{Im} f=F$ iff $f$ is surjective.

Conclusion. For a linear map between spaces of the same finite dimension, injective $\Leftrightarrow$ surjective $\Leftrightarrow$ bijective $\Leftrightarrow$ isomorphism. This fails in infinite dimension (e.g. the shift operator on sequences).

Isomorphism criterion

Statement

Solution

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