Automophism of Vector Space 2

Algebra Level pending

Let $V$ be a finite-dimensional vector space over a field F and there is an surjective linear map $T:V\rightarrow V$ .

Must $T$ be an automophism (isomorphism)?

Bonus: What if finite-dimensional vector space is replaced by free module of finite rank ?

Yes No

1 solution

Otto Bretscher
Nov 6, 2018

The rank-nullity theorem gives $\dim(V)=\dim(\ker T)+\dim(Im T)=\dim(\ker T)+\dim(V)$ so $\ker T=0$ and $T$ is injective. The answer is $\boxed{Yes}$ .

Automophism of Vector Space 2

1 solution

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