TITLE: How do we find the inverse of a vector function with $2$ variables?
QUESTION [9 upvotes]: $$f(m,n) = (2m+n, m+2n)$$
What do we have to do to find the inverse of this function?
I don't even know where to begin.

REPLY [6 votes]: The following is a solution with matrices:
$f$ can be regarded as a linear function, e.g. $\mathbb{Q}^2\to \mathbb{Q}^2$. With respect to the standard basis this is given by the matrix
$$\begin{pmatrix}2&1\\1&2\end{pmatrix}$$
Now you can invert this matrix. There are different methods for that. I'm just using the explicit formula given by Cramer's rule. The inverse is
$$\frac{1}{3}\begin{pmatrix}2&-1\\-1&2\end{pmatrix}$$
Now you can interpret this as a linear function again and get $f^{-1}(m,n)=\left(\frac{2m-n}{3},\frac{-m+2n}{3}\right)$.