elementary set theory - Proving a bijection(injection and surjection) over a function

I need some help proving bijections:

Suppose f is a function from $$\mathbb R^2 \rightarrow \mathbb R^2$$

Defined by

$$f(x,y) = (ax-by,bx+ay)$$

Where a,b are numbers with $$a^2 + b^2 \neq 0$$

Prove that f is a bijection.

I understand that a function f is a bijection if it is both an injection and a surjection so I would need to prove both of those properties.

Could you give me a hint on how to start proving injection and surjection?

Thanks.