Find the standard matrix representation of $T(x,y) = (y,x,x-y)$ 
Find the standard matrix representation of $T: \mathbb{R}^2 \to \mathbb{R}^3$ defined by $T(x,y) = (y, x, x-y)$.

How do I do this problem?
I know that a function that is a linear transformation satisfies 
1) $T(v+u) = T(v) + T(u)$
2) $T(cu) = cT(u)$, for any scalar $c$.
How do I find the standard matrix?
 A: You will need to find a matrix $\begin{bmatrix}a_{1,1}&a_{1,2}\\a_{2,1}&a_{2,2}\\a_{3,1}&a_{3,2}\end{bmatrix}$ such that
$\begin{bmatrix}a_{1,1}&a_{1,2}\\a_{2,1}&a_{2,2}\\a_{3,1}&a_{3,2}\end{bmatrix}\begin{bmatrix}x\\y\end{bmatrix}=\begin{bmatrix}y\\x\\x-y\end{bmatrix}$
Well... let's look at this.  What does it mean for the resulting first row of the product to be equal to the first row of the desired result?

It means that $a_{1,1}x + a_{1,2}y = y$ for all $x,y$.

Clearly, that happens precisely when:

$a_{1,1}=0$ and $a_{1,2}=1$

Indeed, once you see what is going on you should be able to do this quite quickly.

 Notice that $\begin{bmatrix}y\\x\\x-y\end{bmatrix}=\begin{bmatrix}0x+1y\\1x+0y\\1x-1y\end{bmatrix}$... now, look at the coefficients

A: The standard matrix has columns that are the images of the vectors of the standard basis
 $$
T \Bigg (\begin{bmatrix}1\\0\end{bmatrix} \Bigg)=\begin{bmatrix}0\\1\\1\end{bmatrix},  
\qquad
T \Bigg (\begin{bmatrix} 0\\1 \end{bmatrix} \Bigg)=\begin{bmatrix}1\\0\\-1\end{bmatrix}.
$$
So the standard matrix is 

$$\begin{bmatrix}0& 1\\ 1& 0\\1& -1\end{bmatrix}$$

A: You put the images of the basis vectors in columns.
The standard basis vectors $(1,0)\mapsto(0,1,1)$ and $(0,1)\mapsto(1,0,-1)$.
Therefore, the standard matrix is $\pmatrix{0&&1\\1&&0\\1&&-1}$.
You can check that $\pmatrix{0&&1\\1&&0\\1&&-1}\pmatrix{x\\y}=\pmatrix{y\\x\\x-y}.$
A: If by the "standard matrix representation", you meant the matrix of $T$ w.r.t the standard basis of $\mathbb{R}^2$ and $\mathbb{R}^3$. Then just apply $T$ to the standard basis of $\mathbb{R}^2$ and write the image as a linear combination of the standard basis of $\mathbb{R}^3$.
