Construct an isomorphism between $V$ and $\Bbb F^2$ and justify Consider the subspace of $M_2(\Bbb F):V = \biggl\{\begin{pmatrix}a+b&a\\0&b\end{pmatrix}\Biggm\vert  a,b\in \Bbb F\biggr\}$
Construct an isomorphism between $V$ and $\Bbb F^2$ and justify

I assume $F$ is a general field and $M_2(F)$ is the set og $2 \times 2$ matrices under that field. An isomorphism $T: F^2 \rightarrow V$ could simply be the function that maps a point $(a,b) \in F^2$ to the associated matrix in $V$, i.e.
$$T(a,b) = \begin{bmatrix} 
a+b & a \\
0 & b \end{bmatrix}.$$
It is clear that $T$ is a linear tranformation: Let $(a,b), (\alpha, \beta) \in F^2$ and $x, y \in F$. We have
$$T(x(a,b)+y(\alpha, \beta)) = \begin{bmatrix} 
xa + y\alpha + xb + y\beta & xa+ y\alpha \\
0 & xb + y\beta \end{bmatrix} = xT(a,b)+yT(\alpha, \beta)$$.
We can also show that $T$ is injective by assuming $T(a,b) = T(\alpha, \beta)$. We get
$$\begin{bmatrix} 
a+b & a \\
0 & b \end{bmatrix} = \begin{bmatrix} 
\alpha+\beta & \alpha \\
0 & \beta \end{bmatrix} \implies a=\alpha \wedge b=\beta$$
It is also clear that $T$ is surjective by construction. Thus, $T$ is an isomorphism between $V$ and $F^2$.
I may have misinterpreted the question. I hope this could be of some help anyway
