Q:

I can't find the vectors that generate this vectorial subspace

So I have this:
$$ F = \{(x_1, x_2, x_3) : x_1 - x_2 + x_3 = 0\} $$
I'm having trouble finding out the vectors whose linear combination generate the given subspace...
Basically I need something like: 
$x_1(a_1,b_1,c_1) + x_2(a_2,b_2,c_2) + x_3(a_3,b_3,c_3)$ so that given any value for $x_1, x_2$ and $x_3$ the equation gives me a vector in $F$ and so that each of the vectors above is in $F$
I can't find any ...

A:

Note that if you know $x_1$ and $x_2$, then $x_3$ is uniquely determined.  (So, this is a two-dimensional space.)
You can write $x_3=x_2-x_1$.  Therefore, any such vector can be written as
$$
\begin{bmatrix}x_1\\x_2\\x_3\end{bmatrix}=\begin{bmatrix}x_1\\x_2\\x_2-x_1\end{bmatrix}=\begin{bmatrix}x_1\\0\\-x_1\end{bmatrix}+\begin{bmatrix}0\\x_2\\x_2\end{bmatrix}=x_1\begin{bmatrix}1\\0\\-1\end{bmatrix}+x_2\begin{bmatrix}0\\1\\1\end{bmatrix}
$$
So, a reasonable choice of basis is the two vectors $\langle 1,0,-1\rangle^T$ and $\langle0,1,1\rangle^T$.

