Find the image of a vector by using the standard matrix (for the linear transformation T) Was wondering if anyone can help out with the following problem:

Use the standard matrix for the linear transformation $T$ to find the image of the vector $\mathbf{v}$, where
  $$T(x,y) = (x+y,x-y, 2x,2y),\qquad \mathbf{v}=(3,-3).$$

I found out the standard matrix for $T$ to be:
$$\begin{bmatrix}1&1\\1&-1\\2&0\\0&2\end{bmatrix}$$
From here I honestly don't know how to find the "image of the vector $\mathbf{v}$". Does anyone have any suggestions?

I would like to do it in a systematic way. It is instructive to see how OP fits in the following steps.
Let $V$ (resp. $W$) be an $n$ (resp. $m$) dimensional vector space over $\mathbb{R}$. Let $$\alpha=(v_1,\cdots,v_n)$$ be an ordered basis in $V$ and $$\beta=(w_1,\cdots,w_m)$$ an ordered basis in $W$. For any vector $x\in V$, denote its coordinate w.r.t. the basis $\alpha$ as
$$
[x]_\alpha=(x_1,\cdots,x_n)^T
$$
and for any vector $y\in W$, denote its coordinate w.r.t. the basis $\beta$ as
$$
[y]_\beta=(y_1,\cdots,y_m)^T.
$$
Let $T:V\to W$ be a linear transformation. Let $[T]_\alpha^\beta$ denotes the matrix for $T$ w.r.t. the bases $\alpha$ and $\beta$, i.e.,
$$
[T]^\alpha_\beta=[[Tv_1]_\beta,\cdots,[Tv_n]_\beta].
$$
Note in particular that $[T]^\alpha_\beta$ is an $m\times n$ matrix. 
Given $x\in V$, we have
$$
[Tx]_\beta=[T(x_1v_1+\cdots+x_nv_n )]_\beta\\
=x_1[T(v_1)]_\beta+\cdots x_n[T(v_n)]_\beta\\
=[T]_\beta^\alpha[x]_\alpha
$$
