# Thread: Proving that a transformation is surjective (or not)

1. ## Proving that a transformation is surjective (or not)

I've got two questions on a handout that we didn't cover in class...

I guess I'm not understanding what it means to be surjective..

on 5a, would the first blanks be Let Av be in the codomain R^2?

https://gyazo.com/daeb403401c00c164ecb7b876a03a1fc

2. ## Re: Proving that a transformation is surjective (or not)

Originally Posted by MrJank
I've got two questions on a handout that we didn't cover in class...
I guess I'm not understanding what it means to be surjective..
Surjective is a fancy word for onto.
So for any point $\displaystyle \left[ {\begin{array}{*{20}{c}} a\\ b \end{array}} \right]$ you must show that there is a point $\displaystyle \left[ {\begin{array}{*{20}{c}} x\\ y \end{array}} \right]$ such that $\displaystyle T_A\left[ {\begin{array}{*{20}{c}} x\\ y \end{array}} \right]$=$\displaystyle \left[ {\begin{array}{*{20}{c}} a\\ b \end{array}} \right]$

3. ## Re: Proving that a transformation is surjective (or not)

Originally Posted by Plato
Surjective is a fancy word for onto.
So for any point $\displaystyle \left[ {\begin{array}{*{20}{c}} a\\ b \end{array}} \right]$ you must show that there is a point $\displaystyle \left[ {\begin{array}{*{20}{c}} x\\ y \end{array}} \right]$ such that $\displaystyle T_A\left[ {\begin{array}{*{20}{c}} x\\ y \end{array}} \right]$=$\displaystyle \left[ {\begin{array}{*{20}{c}} a\\ b \end{array}} \right]$
Okay, so in the context of my problem what would the beginning of a proof look like?

4. ## Re: Proving that a transformation is surjective (or not)

Originally Posted by MrJank
Okay, so in the context of my problem what would the beginning of a proof look like?
Let $\vec{v}$ be in the codomain of $T_A$.

Or

Let $\vec v$ be in the codomain $\mathbb{R}^2$

5. ## Re: Proving that a transformation is surjective (or not)

Originally Posted by MrJank
Okay, so in the context of my problem what would the beginning of a proof look like?
$\displaystyle T_A\left[ {\begin{array}{*{20}{c}} x\\ y \end{array}} \right]=\left[ {\begin{array}{*{20}{rr}} 2x+y\\ -4x+2y \end{array}} \right]=\left[ {\begin{array}{*{20}{c}} a\\ b \end{array}} \right]$
Now YOU solve for $x~\&~y$. Show us your work!