Automophism of Vector Space

Algebra Level 3

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

Must $T$ be an automophism (isomorphism)?

Bonus: What if finite-dimensional is replaced by infinite-dimensional ?

No Yes

1 solution

Otto Bretscher
Oct 11, 2018

If $V$ is finite-dimensional, then $T$ will be onto, by the rank-nullity theorem.

If $V$ is infinite-dimensional, then $T$ may not be an automorphism. As a counter-example, consider $T(f(x))=xf(x)$ on the space of polynomials $\mathbb{R}[x]$ .

Automophism of Vector Space

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