Given a mapping for $\mathbb R^2\rightarrow\mathbb R^3$ , $A,$ find new basis for $\mathbb R^2$ and $\mathbb R^3$ such that the mapping is now diagonal. Note that $\mathbb R^2$ and $\mathbb R^3$ currently use the standard basis.
I was thinking of using singular value decomposition. In this case would the new basis for $\mathbb R^2$ be the columns of $V$ and $\mathbb R^3$ be the columns of $U$ for $A = US$($V$ transpose)