Construct an isomorphism from $2\times 2$ matrix to $\mathbf F^3$

Question

Let $$V=\{\begin{pmatrix} a&a+b\\ 0&c\\ \end{pmatrix} :a, b, c\in\mathbf F\}$$ Construct an isomorphism from $V$ to $\mathbf F^3$

I think I have to find a invertible linear map between $V$ and $(x, y, z)\in\mathbf F^3$

But how to find it?

Answer 1

You can take$$\begin{array}{rccc}\Psi\colon&\mathbf F^3&\longrightarrow&V\\&(a,b,c)&\mapsto&\begin{bmatrix}a&a+b\\0&c\end{bmatrix}.\end{array}$$

Answer 2

Note that $$V=\left\{\begin{pmatrix} a&a+b\\ 0&c\\ \end{pmatrix} \middle| a, b, c\in\mathbf F\right\} = \left\{\begin{pmatrix} a&b\\ 0&c\\ \end{pmatrix} \middle| a, b, c\in\mathbf F\right\}$$ And then it is easy to see this is a $3$ dimensional vector space over $\mathbb{F}$, so a simple isomorphism is $$\varphi \left( \begin{bmatrix} a & b \\ 0 & c \end{bmatrix} \right) = \left(a,b,c\right)$$ Note that all you need to do is find a map that takes basis vectors to basis vectors for two $n$-dimensional vector spaces over a field.