Prove that $f(m,n)=(m+2n, m-n)$ is 1-1 and Onto. The domain and co-domain are $\mathbb R\times \mathbb R$

Question

So I know how to prove injectivity $f(x)=f(y)\Rightarrow x=y$ and surjectivity but am not sure how to go about it in this case since there are multi variables.

Answer 1

Note that $f:\mathbb R^2\rightarrow\mathbb R^2$ is given by $f(x)=Ax$ where $$ A= \begin{pmatrix} 1 & 2 \\ 1 & -1 \end{pmatrix} $$ But $\det A=-1-2=-3\neq0$ so $A$ is invertible. Hence $f$ is one-to-one and onto.

Answer 2

The methods are precisely the same. For injectivity, you assume

$f(m_1,n_1) = f(m_2, n_2)$ and show that this implies $(m_1,n_1) = (m_2,n_2)$. For surjectivity, you take a generic point, say $(x,y)$, and you need to find the choice of $(m,n)$ such that $f(m,n) = (x,y)$.