MTH2021 · Linear Algebra with Applications
Inner Product Spaces
Week 8 adds geometry to vector spaces through an inner product ⟨u, v⟩ satisfying symmetry, linearity and positive definiteness. It induces a norm ‖v‖ = √⟨v, v⟩ and a distance, and gives the Cauchy–Schwarz and triangle inequalities, angles, orthogonality and orthogonal complements. Inner products can be Euclidean, weighted, or integral inner products on function spaces. This material opens the second half of the unit; it is assessed only in the comprehensive 50% final and sets up Gram–Schmidt and least squares.
What this chapter covers
- 01Inner-product axioms: symmetry ⟨u,v⟩ = ⟨v,u⟩, linearity ⟨u, v + kw⟩ = ⟨u,v⟩ + k⟨u,w⟩, positive definiteness ⟨v,v⟩ ≥ 0 with equality ⇔ v = 0
- 02Examples: the dot product on ℝⁿ; weighted ⟨u,v⟩ = Σ wᵢuᵢvᵢ (wᵢ > 0); integral ⟨f,g⟩ = ∫ f g on function spaces
- 03Induced norm ‖v‖ = √⟨v,v⟩ and distance d(u,v) = ‖u − v‖
- 04Cauchy–Schwarz inequality: |⟨u,v⟩| ≤ ‖u‖‖v‖
- 05Triangle inequality ‖u + v‖ ≤ ‖u‖ + ‖v‖ and the norm/metric properties
- 06Angle θ = arccos(⟨u,v⟩ / (‖u‖‖v‖)); orthogonality ⟨u,v⟩ = 0
- 07Pythagoras: u ⊥ v ⇒ ‖u + v‖² = ‖u‖² + ‖v‖²
- 08Orthogonal complements and the move toward projections
A weighted inner product: norms, angle and Cauchy–Schwarz
- +1Inner product: ⟨u, v⟩ = 2·(1)(2) + 5·(2)(3) = 4 + 30 = 34.
- +1Norms from ‖x‖ = √⟨x, x⟩: ‖u‖ = √(2·1² + 5·2²) = √(2 + 20) = √22; ‖v‖ = √(2·2² + 5·3²) = √(8 + 45) = √53.
- +1Cauchy–Schwarz |⟨u,v⟩| ≤ ‖u‖‖v‖: right side = √22·√53 = √1166 ≈ 34.15, and |34| = 34 ≤ 34.15 ✓ (the vectors are nearly parallel in this geometry). The angle is θ = arccos(34/√1166) ≈ 5.1°.
- +1Distance: u − v = (−1, −1), so d(u, v) = ‖u − v‖ = √(2·(−1)² + 5·(−1)²) = √(2 + 5) = √7 ≈ 2.65.
Key terms
- Inner product
- A rule ⟨u, v⟩ that is symmetric, linear in its second slot, and positive definite (⟨v,v⟩ ≥ 0, = 0 only for v = 0); it generalises the dot product.
- Induced norm
- ‖v‖ = √⟨v, v⟩, the length attached to an inner product; distance is d(u, v) = ‖u − v‖.
- Weighted inner product
- ⟨u, v⟩ = Σ wᵢuᵢvᵢ with all wᵢ > 0; a valid inner product on ℝⁿ that rescales the axes, changing lengths and angles.
- Cauchy–Schwarz inequality
- |⟨u, v⟩| ≤ ‖u‖‖v‖, with equality exactly when u and v are parallel; it guarantees the angle formula is well defined.
- Orthogonality
- u and v are orthogonal when ⟨u, v⟩ = 0; then Pythagoras holds: ‖u + v‖² = ‖u‖² + ‖v‖².
- Orthogonal complement
- For a subspace U, the set U^⊥ of all vectors orthogonal to every vector of U; it is itself a subspace and pairs with U in the projection theorem.
Inner Product Spaces FAQ
What makes something an inner product?
Three axioms: symmetry ⟨u,v⟩ = ⟨v,u⟩, linearity in one slot, and positive definiteness (⟨v,v⟩ ≥ 0, equal to 0 only when v = 0). The dot product is the model, but weighted sums Σwᵢuᵢvᵢ (with positive weights) and integrals ∫fg on function spaces also qualify, and each gives its own notion of length and angle.
Why does the norm have to come from the inner product?
Because ‖v‖ = √⟨v, v⟩ is how length is defined in an inner product space — positive definiteness guarantees the quantity under the root is non-negative and zero only for v = 0. If you switch to a weighted or integral inner product, you must compute the norm with the same rule, not with Euclidean length.
What is Cauchy–Schwarz for?
It bounds the inner product by the product of the norms, |⟨u,v⟩| ≤ ‖u‖‖v‖, which is exactly what makes the angle formula θ = arccos(⟨u,v⟩/(‖u‖‖v‖)) well defined (the ratio stays in [−1, 1]). It also proves the triangle inequality and is a classic short-proof target.
How do I know if two vectors are orthogonal?
They are orthogonal precisely when their inner product is zero, ⟨u, v⟩ = 0 — and 'zero' is measured in whatever inner product you are using, so two vectors can be orthogonal under a weighted product but not the Euclidean one. Orthogonality then unlocks Pythagoras and the projection machinery of the next chapter.
Is inner-product material on the tests or only the exam?
Week 8 falls after Test 2's coverage (Weeks 3–7), so inner product spaces are assessed in the weekly quiz for that week and, most importantly, in the comprehensive 50% final. They are also the foundation for Gram–Schmidt, projection and least squares, which the exam leans on heavily.
Exam move
Practise every computation twice — once with the Euclidean dot product and once with a weighted or integral inner product — so you never slip back to ordinary lengths when the geometry has changed. Memorise the axioms and be ready to prove Cauchy–Schwarz and the triangle inequality, since those are common handwritten-exam items. Week 8 is assessed in its quiz and the 50% final and underlies Gram–Schmidt and least squares. When a norm or angle looks off, ask Sia to recompute it in the correct inner product step by step.
Working through Inner Product Spaces in MTH2021? Sia is AskSia’s AI Mathematics tutor — ask any MTH2021 Inner Product Spaces question and get a clear, step-by-step explanation grounded in how MTH2021 is taught and assessed. Read this chapter free, then take your hardest questions to Sia.
