I have the two following questions that I'm having trouble approaching. In what way would I find an isomorphism here?
Define V and G:
$V=\big\{x=\begin{pmatrix}x_1 \\ x_2 \\ x_3 \end{pmatrix}\in \mathbb{R}^3; x_1+x_2+x_3=0\big\}$
$W=\big\{x=\begin{pmatrix}x_1 \\ x_2 \\ x_3 \end{pmatrix}\in \mathbb{R}^3; x_3=0\big\}$
Give an isomorphism $g:V\rightarrow W$ such that $g(\begin{pmatrix}1 \\ -1 \\ 0 \end{pmatrix})=\begin{pmatrix}1 \\ 0 \\ 0 \end{pmatrix}$ and $g(\begin{pmatrix} 1 \\ 0 \\ -1 \end{pmatrix})=\begin{pmatrix}0 \\ 1 \\ 0 \end{pmatrix}$.
What is $g(\begin{pmatrix}x_1 \\ x_2 \\ x_3 \end{pmatrix})$ for $\begin{pmatrix}x_1 \\ x_2 \\ x_3 \end{pmatrix}\in V$?
To me it looks like there is something going on with the identity matrix, but I don't know where to start. Any pointers would be greatly appreciated.
Write $v_1 = (1,-1,0)^T$ and $v_2 = (1,0,-1)^T$. Then $v = (x_1,x_2,x_3)^T$ can be written as
$$ v = -x_2 v_1 - x_3 v_2.$$
Using the linearity of $g$, we have
$$ g \begin{pmatrix} x_1 \\ x_2 \\ x_3 \end{pmatrix} = g(-x_2 v_1 - x_3 v_2) = -x_2 g(v_1) - x_3 g(v_2) = \begin{pmatrix} -x_2 \\ 0 \\ 0 \end{pmatrix} - \begin{pmatrix} 0 \\ -x_3 \\ 0 \end{pmatrix} = \begin{pmatrix} -x_2 \\ -x_3 \\ 0 \end{pmatrix}.$$