Dimensions of linear transformation in relation to kernel

Question

I am new to linear algebra, and I just wanted to doublecheck my understanding of the following:

$T: R^n \to R^m$ is a linear transformation.

True or false?

If $n>m$, then $\operatorname{Ker}T \neq {0}$.

This statement is correct, because if $n>m$, then per definition $T$ is not injective. If $T$ is not injective, then $ \operatorname{Ker}T\neq {0}$.

Is my understanding correct?

Thank you!

Answer 1

It is true.

The rank nullity theorem says $\operatorname{rank}T+\operatorname{nullity}T=n$.

On the other hand, $\operatorname{rank}T\le m\lt n$. $\therefore \operatorname{nullity}T\gt0$.