Let V = $\begin{Bmatrix} a&a+b\\ 0&c \end{Bmatrix}$ where a, b and c are in F, and construct an isomorphism from V to $F^3$.
I let T(V) = $(a,b,c)$ or $(a,d,c)$, where $d=a+b$. I know this spans all of $\Bbb{R}^3$, is linear, 1-1 and onto. I proved linearity, but I really don't understand 1-1 / onto proofs and my textbook doesn't provide much help.
If your space $V$ is $$ V=\left\{\begin{bmatrix} a & a+b \\ 0 & c\end{bmatrix} \biggm| a,b,c\in F\right\} $$ which is a subspace of the vector space of $2\times2$ matrices over $F$, you can easily find a basis, because $$ \begin{bmatrix} a & a+b \\ 0 & c\end{bmatrix} = a\begin{bmatrix} 1 & 1 \\ 0 & 0\end{bmatrix}+ b\begin{bmatrix} 0 & 1 \\ 0 & 0\end{bmatrix} c\begin{bmatrix} 0 & 0 \\ 0 & 1\end{bmatrix} $$ Send the first matrix in the basis to $(1,0,0)$, the second to $(0,1,0)$ and the third to $(0,0,1)$. So $$ f\colon\begin{bmatrix} a & a+b \\ 0 & c\end{bmatrix} \mapsto (a,b,c) $$ and this is an isomorphism by construction.