I would like to show the following claim:
Let $l_1, l_2 \in \mathbb{R}^{n+1}$ be two distinct lines then $\exists$ a linear isomorphism $A : \mathbb{R}^{n+1} \rightarrow \mathbb{R}^{n+1}$ with $Al_1 = (\lambda, 0, 0, \dots , 0)$ and $Al_2 = (0, \lambda, 0, \dots , 0)$
I thought this $A$ needed to prove the claim is the basis transformation matrix from the standard basis of $\mathbb{R}^{n+1}$ into the basis consisting of $l_1$, $l_2$ and some linearly independent basis vectors.
My question now is: Are my thoughts correct? And if so, how can I construct the missing vectors to make $\{l_1, l_2 \}$ into a basis of dimension $n + 1$?