Determine a linear application How can I determine a linear application $f: \mathbb{R}^2 \to \mathbb{R}^2 $ such that $Kerf = span[ (-1, -5) $? By the definition of Ker, we know that $ kerf = [ (x,y) \in \mathbb{R}^2 : f(x,y) = (0,0)] $. That is, we need to find a linear application such that f applied in the vectors that are linear combination of the vectors that form one basis of the kernel, we will have the 0 vector. But is there a method to find one application like this? I need a hint, probably I'm not seeing the best way to determine one application Thanks!

Hint: try to find a linear map $\Bbb R^2\to\Bbb R$ with that kernel. Then to make an example $\Bbb R^2\to\Bbb R^2$, you can compose with a (almost any) linear map $\Bbb R\to\Bbb R^2$ so that the composition has the same kernel.