Inner-product spaces
Objective. Use orthogonality, projections and Gram-Schmidt.
- 1. Learn
- 2. Worked example
- 3. Practice
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Learn: the key idea
Inner-product spaces equip a vector space with a map ⟨·,·⟩ satisfying linearity, symmetry (or conjugate symmetry) and positive-definiteness, which induces a norm ‖v‖ = √⟨v,v⟩ and notion of orthogonality. The Gram-Schmidt process converts any basis {v₁,…,vₙ} into an orthonormal basis by successively subtracting projections onto previous vectors: uₖ = vₖ − Σᵢ projᵤᵢ(vₖ). Projection onto a subspace minimises distance to that subspace, key for least-squares problems. For v₁ = (1,1), v₂ = (1,0) in ℝ² with the standard inner product, Gram-Schmidt gives u₁ = (1,1)/√2 and u₂ orthogonal to u₁, found by subtracting the projection of v₂ onto u₁.
Worked example
Apply Gram-Schmidt to the vectors v₁ = (1,0) and v₂ = (1,1) to find an orthonormal basis for ℝ².
- 1Set u₁ = v₁ = (1,0), already of unit length since ‖u₁‖ = 1.
- 2Compute the projection of v₂ onto u₁: proj = ⟨v₂,u₁⟩u₁ = (1)(1,0) = (1,0).
- 3Subtract to get the orthogonal component: w₂ = v₂ − proj = (1,1) − (1,0) = (0,1).
- 4Normalise w₂: since ‖w₂‖ = 1, the orthonormal basis is {(1,0), (0,1)}.
Answer: {(1, 0), (0, 1)}
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Start practisingCommon mistake. Students normalise vectors before subtracting projections rather than after, or forget to normalise at all, leaving an orthogonal but not orthonormal basis when an orthonormal one was required.
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