Inverse function in $\mathbb R^2$

Question

How do I find the inverse function of $f: X\to Y$ where $X,Y$ both are subsets of $\mathbb R^2$ and $f$ is defined as $f(x,y)=(x+y,x-y)$.

Answer 1

Hint:$$\underbrace{\begin{bmatrix}x \\y \end{bmatrix}}_f\mapsto\begin{bmatrix}1 &1 \\1 & -1 \end{bmatrix}\begin{bmatrix}x \\y \end{bmatrix}$$ find inverse of matrix to find $f^{-1}$

Answer 2

So this is a system of linear equations, so the core thing is to solve the system. For larger degree inputs and outputs, matrix inversion is the way to go, but for this case, solving the equations without them isn't too horrible:

$f(x, y) = (u, v)$ -> $f^{-1} (u, v) = (x, y)$

$x + y = u$

$x - y = v$

$u + v = 2x + 0 = 2x$

$x = \frac{u + v}{2}$

$y = u - x = u - \frac{u + v}{2} = \frac{u - v}{2}$

$f^{-1} (u, v) = (\frac{u + v}{2}, \frac{u - v}{2})$

Answer 3

Solve for $(x,y)=g(u,v)$ the following system

$$\begin{align} u=&x+y\\ v=&x-y \end{align}$$

You find that

$$\begin{align} x={u+v\over 2}\\ y={u-v\over 2} \end{align}$$

Therefore

$$g(u,v)=f^{-1}(u,v)=\left({u+v\over 2},{u-v\over 2}\right)$$