• To express as a linear combination of our basis vectors, we start with:
  • Normally we would write out the equations and do RREF.
  • Here we multiply both sides by , it’s just an arbitrary vector in the orthogonal set.
  • Distributing the dot product to all the elements of the LI equation, we see that all the terms except one become zero.
  • Since is orthogonal to every vector in the set except , all the other dot products zero out.

Just for fun:

Makes sense too, the vector is the sum of it’s components projected onto the basis vectors.