Let V be vector space in Rˆn.
Using Gram-Schmidt's orthogonalization procedure notations, show that:
a) if j >= 2, v_j is perpendicular to v_i for every i < j and v_j != 0.
b) the set C = { v_1, ... , v_n) is linearly independent
Conclude that C is an orthogonal basis of V.
My doubts are:
The exercise does not ask to prove the general procedure itself, right? It only asks to show a sort of particular case, when j >= 2. I've seen proofs using induction. Is this the right approach to the problem?
Once I show a), does it not automatically implies b) and the conclusion asked?