I don't know really, all I want is to now find the vector v' cords.
Actually v == v', they are the same point in space. v is defined as (-0.5, 0, -1), but I want its coords in therms of matrix M.
Imagine it like this, at the beginning we are located at origin of our coord system, we look around and we see another object (also at origin) with its local coord system as M. Its rotated a little bit counter clockwise (right hand system). And then you also see a point in space located at v (-0.5, 0, -1).
Now you transfer to the object coord system, and suddenly the matrix M that you saw before becomes:
(1, 0, 0)
(0, 1, 0)
(0, 0, 1)
And also everything else changes (coords of all objects) because we now look at everything from another coord system. But all objects and points haven't moved. They are at the same place as before. Now you take a look a point v and you see it at same position as before, that's why v == v'.
So now I want to know the coords of point v', that is from the point of view of our object (that rotated a little bit).