Quick confirmation

**mathswannabe** · September 8th 2011, 02:43 AM

Hi All, this shouldn't take anyone worth their salt too long.... i'm just wondering if this is correct:

If the task is to prove that:

$(u,v)\rightarrow (u+v,u-v)$ is a bijection from $\mathbb{R}^2$ onto $\mathbb{R}^2$ ...

Do we go about it this way?

To show injectivity:

Assume there exists $(u_1,v_1)\neq(u_2,v_2)$ such that: $u_1+v_1 = u_2 + v_2,$ and also that $u_1 - v_1 = u_2 - v_2$ .

Isolating $u_1$ in the first equation, we can sub in to the second to get $u_1=u_2$ and then we can repeat the similar process, just isolating $v_1$ to arrive at $v_1=v_2$ ?

Then to show surjectivity:

Do we just pick any $(x,y)$ in $\mathbb{R}^2$ . We set $(x,y)=(u+v,u-v)$ and then solve for $u$ and $v$ in terms of $x$ and $y$ . I get $u=\frac{x+y}{2}$ and $v=\frac{x-y}{2}$ ??