Im(T) Isn't Onto if Rows Of A> Columns Of A

Question

Let there be a linear map and its corresponding matrix $A\in \mathbb{F}^{mxn}$.

is it ok to say that Looking at $Ax=b$, If $m>n$ so the linear map can not be onto?

Answer 1

Your matrix $A$ corresponds to a linear transformation $T : \mathbb{F}^n \to \mathbb{F}^m$. The image of $T$, call it $Im(T)$, is a subspace of $\mathbb{F}^m$ with dimension at most $n$, since the image of a basis for $\mathbb{F}^n$ will be a generating set for the image $Im(T)$. Since $m > n$, $Im(T)$ is a proper subspace of $\mathbb{F}^m$, and so $T$ is not onto.