How to check if a subset is a generator of a vector space I have a very noob questions about generators: what algorithm do I have to follow so I can prove that a finite subset is a generator?
Here is the background story (I'll tell it all because I suck at maths and I might have understood the whole concept wrongly):
I want to know how to compute the image of a vector space morphism. I'll continue on a concrete example:
$f\colon \mathbb{R}^2 \to \mathbb{R}^2$, $f(x,y) = (x-y, 2x + y)$ is the morphism.
To find its image, I observe that $f(x,y)$ can be rewritten as $x(1,2) + y(-1,1)$. If I am correct, that means that $\mathrm{Im}(f) = \langle(1,2), (-1,1)\rangle$ (that's the notation we use for set generators). Now the question is: does the $\{(1,2), (-1,1)\}$ set generate $\mathbb{R}^2$ or not?

If you want to know whether the two vectors span the plane, just use the standard results in this case: they span the plane if and only if they are linearly independent, or the $2$-by-$2$ matrix is invertible, and so on.
Alternatively you can just show that for every point in the plane there exists the required $x$ and $y$ by solving a pair of simultaneous equations. 
