Inner products and orthonormal bases
General · Mathematics
Study notes
Q: Expand (3,4) in orthonormal basis e₁ = (1/√2)(1,1), e₂ = (1/√2)(1,-1). c₁ = ⟨(3,4),e₁⟩ = (3+4)/√2 = 7/√2. c₂ = ⟨(3,4),e₂⟩ = (3-4)/√2 = -1/√2. (3,4) = (7/√2)e₁-(1/√2)e₂. Check: (7/2-1/2, 7/2+1/2) = (3,4) ✓! Parseval: 9+16 = 49/2+1/2 = 25 ✓!