Show that if $W$ is a finite dimensional $K$-vector space any linear surjective map $f:W\to W$ is bijective.
I feel that the rank nullity theorem is needed for this one…
We are given that $f$ is surjective, so we must show that it is also injective, then we have bijectivity. But how would I show $f$ is injective when $f$ is a general mapping?
Best Answer
Yes, use the rank-nullity theorem, also called the dimensions' theorem:
$$\dim W=\dim\ker T+\dim\text{Im}\,T\stackrel{\text{given}}=\dim\ker T+\dim W\implies\dim\ker T=0$$
Now just use/prove that any linear map is injective iff $\;\ker T=\{0\}\;$ .