I want to know the coord of vector v (-0.5, 0, -1) from the object (M) point of view.
I haven't sad that v' = Mv. I sad that I just multiplied it like that to get some point (to se what Ill get). But that's wrong because that's rotated point, and point v doesn't rotate and move. No point in space or object moves or rotates.
I simply transfer my point at "looking" at the world. The matrix M becomes my original point of view. Because of this every coordinate changes, even of point v.
The original coord system, the one where everything was defined, also changes and its right, up, forward coords when looked from new object (M).
Point v has not changed its position in terms where its located it space physically, just its coords change because we look at it from another point (M).