Change of Basis: One Vector, Two Sets of Coordinates

The black arrow x stays fixed for the whole film. In the grey original basis its coordinates are one, one half, and one: the grey path walks along e zero, e one and e two to its tip.

Here is a new basis: the red e zero prime, the blue e one prime, and the green e two prime. Their coordinates in the old basis are the three columns of the matrix V.

The same x is also a walk along the new arrows: z zero steps of red, z one of blue, z two of green. So x equals V times z, and z is x written in the new basis.

Now the new arrows tilt. The black x does not move at all. Only z changes, because the ruler changed, not the vector.

When the new arrows have length one and meet at right angles, V is orthonormal, and z is simply V transpose times x.

Your turn. Drag the tip of the red arrow until it points exactly along x, and watch z.

Along x, the blue and green steps vanish, and z has a single nonzero entry. Move any new arrow, or the numbers of V, and the walk follows.