If $V$ and $W$ are both n-dimensional vector spaces and $L : V → W$ is a linear mapping, then $\text{nullity}(L) = 0$.

Question

Question: If $V$ and $W$ are both n-dimensional vector spaces and $L : V \longrightarrow W$ is a linear mapping, then $\text{nullity}(L) = 0$.

This is a true or false question but Im not sure which way is correct.

Can someone point me in the right direction?

I know it has something to do with the Rank-Nullity Theorem, and the question is essentially asking of the dimension of the Range of $L$ is equal to $n$, but I'm not sure how to get there.

Answer 1

This is false. For any pair of $n-$dimensional vector spaces $V,W$ there is the zero map; that is a map $f:V\longrightarrow W$ such that $f(v)=0_W$ for any $v\in V$. It is very easy to see that such map is linear and $null(f)=n$.