# Proving range of transformation

1. Oct 6, 2014

### jonroberts74

1. The problem statement, all variables and given/known data
I haven't learned kernel yet so if thats of use here I don't know it yet

let $T: \mathbb{R^3} \rightarrow \mathbb{R^2}$ where $T<x,y,z>=<2y,x+y+z>$

prove that the range is $\mathbb{R^2}$

3. The attempt at a solution
I know that T is not one-to-one, I checked that for a previous question. I did read if the range is $\mathbb{R^n}$ then T is onto. but I am not really sure how to prove the range is $\mathbb{R^2}$

2. Oct 6, 2014

### RUber

A good way to do this is to choose any (a,b) in $\mathbb{R}^2$. Since they are arbitrary points in the range, if you can find a general form for them which is in the domain (in this case $\mathbb{R}^3$), then the range is the entirety of $\mathbb{R}^2$.

3. Oct 6, 2014

### jonroberts74

so it goes both way "if $T$ is onto $\Leftarrow\Rightarrow$ the range is $\mathbb{R^n}$"

thanks

I think I was misunderstanding what range meant in this sense.

4. Oct 7, 2014

### Staff: Mentor

Since T is a transformation to $\mathbb{R^2}$ what you said should refer to that space, not $\mathbb{R^n}$.