I don't know how to find coordinate systems for a question.

Question

MST is $3 \times 3$ homogeneous matrix that transform points from coordinate system $S$ to coordinate system $T$.

1- What are the coordinates of $P$ in coordinate system $O$?

2- What are the coordinates of $P$ in coordinate system $A$?

3- What are the coordinates of $P$ in coordinate system $B$?

4- What is the matrix of MAB? Please show a bit detail of derivation for this i'd be happy?

Answer 1

To get to $P$ from the origin of $O$ you go $3$ units in the $x$ direction and $1$ unit in the $y$ direction. Therefore the coordinates of $P$ in $O$ are $(3, 1).$

To get to $P$ from the origin of $A$ you go $1$ unit in the $x$ direction and $½$ unit in the $y$ direction. Therefore the coordinates of $P$ in $A$ are $(1, ½).$

To get to $P$ from the origin of $B$ you go $½$ unit in the $x$ direction and $-½$ unit in the $y$ direction. Therefore the coordinates of $P$ in $B$ are $(½, -½).$

I don't know the exact definition of $M_{AB}$ and can therefore not help much with that. I suppose that it contains both a transform of the axis directions and of the origin. But exactly how?