Prove that $T$ is an isomorphism $\iff \ker(T)=\{0\}$

Question

Prove that $T$ is an isomorphism $\iff \ker(T)=\{0\}$

I'm not able to prove this statement. Can you please help me with the proof.

Answer 1

As it's written it's not true. Take $T:\mathbb R^2\to \mathbb R^3$ defined by $T(x,y)=(x,y,0)$. It's injective (i.e. $\ker T=\{0\}$) but it's not an isomorphism.

The statement should be : let $T:V\to W$ linear with $\dim(V)=\dim(W)$. Then $T$ is an isomorphism $\iff$ $\ker(T)=\{0\}$. And this is an obvious consequence of Rank-Nullity theorem.