Linear Algebra One to one and onto function

Question

I was just wondering how I can tell if a function is onto. $\mathbf{R}^3\to\mathbf{R}^1$ Lets say the standard transformation matrix has vectors $\{1,0,0\}$, $\{0,1,0\}$, $\{0,0,0\}$. I know that this transformation is not one to one since they vectors are not linearly independent. How can I see if the transformation is onto?

Answer 1

For a linear transformation $T$ from $\mathbb{R}^n$ to $\mathbb{R}^n$, the following are equivalent:
(1) $T$ is one-one
(2) $T$ is onto
(3) If $T(v)=0$, then $v=0$

Generally, for a linear transformation $T: \mathbb{R}^n \longrightarrow \mathbb{R}^m$, the following are equivalent:
(1) $T$ is onto
(2) There is some basis $\mathcal{B}$ of $\mathbb{R}^m$ such that the image of $T$ contains $\mathcal{B}$

Answer 2

I generally try to use the Rank-Nullity theorem whenever I can.

The Rank Nullity Theorem. Let $f:V\rightarrow W$ be a linear map between finite-dimensional vector spaces $V$ and $W$. Then $$ \operatorname{rk}f+\operatorname{nul}f=\dim V $$ where $\operatorname{rk}f:=\dim\operatorname{im}f$ and $\operatorname{nul}f:=\dim\ker f$.

So, to check if a linear map $f:\mathbb R^m\rightarrow\mathbb R^n$ is onto, we need only check that $\operatorname{nul}f=m-n$.