How to find the matrix that represents a linear transformation? I'm given the exercise:

Given the subspace: 
  $$U =\left\{ \textit{x} = \left(\begin{matrix} x_1 \\ x_2 \\ x_3 \end{matrix} \right) \in \mathbb{F} | x_1=x_2+x_3 \right\} \subset \mathbb{F}^3$$ 
  and the function $f: U \rightarrow \mathbb{F}^2$ defined by: 
  $$f \left(\begin{matrix} x_1 \\ x_2 \\ x_3 \end{matrix} \right) = \left(\begin{matrix} x_1 \\ x_3 \end{matrix} \right) $$ 
  a) Find a basis for $U$ 
  b) Find the matrix $A$ that represents the linear transformation with respect to the found basis of $U$ and the standard basis  $(e_1,e_2)$ for $\mathbb{F}^2$ 
  c) Show that $f: U \rightarrow \mathbb(F)^2$ is isomorphic

I have found the basis of $U$ to be $\left(\left(\begin{matrix} 1 \\ 1 \\ 0 \end{matrix} \right) , \left(\begin{matrix} 1 \\ 0 \\ 1 \end{matrix} \right)\right)$. However I have no idea how I should interpret or solve question b. Question c shouldn't be a problem when I've solved b. Any help is much appreciated. 
 A: Here is the more general setting. 
Let $V$ (resp. $W$) be an $n$ (resp. $m$) dimensional vector space over $\mathbb{F}$. Let $\alpha=(v_1,\cdots,v_n)$ be an ordered basis in $V$ and $\beta=(w_1,\cdots,w_m)$ an ordered basis in $W$. 
In your question, $V=U$ and $W=\mathbb{F}^3$. You have found $\alpha$. Note that $\beta$ is given. Also, $n=2$ and $m=2$.
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.,
  $$
[T]^\alpha_\beta=\bigr[[Tv_1]_\beta,\cdots,[Tv_n]_\beta\bigr].
$$
  Note in particular that $[T]^\alpha_\beta$ is an $m\times n$ matrix. 


So what are the steps to find $[T]^\alpha_\beta$?


*

*Note that you are given $\alpha$ and $\beta$, and $T$. Identify what are $m$ and $n$ for your problem;

*Find $Tv_j$ for each $j$;

*and then find $[Tv_j]_\beta$.


For the very last step, you need to know how to find $[z]_{\beta}$ given $z\in W$.  Suppose $[z]_\beta=(z_1,\cdots,z_m)^T$ Then by the definition of coordinates:
$$
z=\sum_1^m z_iw_i
$$
which essentially gives you a linear equation about the $z_i$'s.
