The question is:
In $\mathbb{R}^4$ plane $V$ is given, $V=\mathrm{span}(\alpha_1,\alpha_2)$ where $\alpha_1=[1,3,4,1]$, $\alpha_2=[1,2,2,3] $
a) Find the formula for isomorphism $\varphi:\mathbb{R}^4\rightarrow\mathbb{R}^4$, such that $\varphi(V)=\mathrm{span}(\epsilon_1,\epsilon_2)$
b) Find the system of equations which will describe $V$.
I know how to solve a) but have problems with solving b) I am looking for straightforward solutions. (i've posted subpoint a) because I think it is somehow related to b))
It seems the following.
You intuition is right, because $\varphi(V)$ is the orthogonal complement to $\mathrm{span}(\epsilon_3,\epsilon_4)$. So the required system is
$\cases{ (\varphi(v), \epsilon_3)=0\\ (\varphi(v), \epsilon_4)=0 }$