4.1 Inner products and their properties

Inner products are fundamental to understanding vector spaces more deeply. They allow us to measure angles, lengths, and distances between vectors, giving us powerful tools for geometric analysis and computation.

These mathematical objects form the foundation of inner product spaces, a crucial concept in linear algebra. They lead to important results like the Cauchy-Schwarz inequality and enable us to define orthogonality, which has wide-ranging applications in mathematics and physics.

Inner Products and Properties

Definition and Key Properties

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• Inner product maps V × V to F in vector space V over field F
• Satisfies conjugate symmetry, linearity in first argument, and positive definiteness
• For real vector spaces, conjugate symmetry becomes $⟨x, y⟩ = ⟨y, x⟩$ for all x, y ∈ V
• Linearity in first argument means $⟨αx + βy, z⟩ = α⟨x, z⟩ + β⟨y, z⟩$ for all x, y, z ∈ V and α, β ∈ F
• Positive definiteness ensures $⟨x, x⟩ ≥ 0$ for all x ∈ V, with equality only when x = 0
• Examples of inner products
  • Dot product in R^n
  • Complex inner product in C^n

Induced Norm and Orthogonality

• Inner products induce norm on vector space defined as $||x|| = √⟨x, x⟩$
• Induced norm satisfies all properties of a norm (non-negativity, positive definiteness, homogeneity, triangle inequality)
• Orthogonality between vectors defined using inner product
  • x and y are orthogonal if $⟨x, y⟩ = 0$
• Examples of orthogonal vectors
  • (1, 0) and (0, 1) in R^2
  • sin(x) and cos(x) in function space C[0, 2π]

Cauchy-Schwarz Inequality

Statement and Proof

• Cauchy-Schwarz inequality states $|⟨x, y⟩|² ≤ ⟨x, x⟩⟨y, y⟩$ for all x, y in inner product space
• Proof typically involves quadratic function $f(t) = ⟨x + ty, x + ty⟩$
  • Show f(t) is non-negative for all real t
• Equality case occurs if and only if x and y are linearly dependent
• Examples demonstrating inequality
  • In R^2: |(2, 3) · (1, -1)|² ≤ (2² + 3²)(1² + (-1)²)
  • For complex numbers: |z₁z₂*| ≤ |z₁||z₂|

Applications and Consequences

• Leads to triangle inequality for induced norm: $||x + y|| ≤ ||x|| + ||y||$
• Used for bounding inner products and proving inequalities in analysis
• Establishes relationships between different norms
• Crucial in proving continuity of inner product and norm functions
• Defines angle between vectors: $cos θ = ⟨x, y⟩ / (||x|| ||y||)$
• Applications in signal processing and information theory
  • Bounding correlation between signals
  • Deriving capacity of communication channels

Inner Product Spaces

Standard Examples

• R^n with dot product: $⟨x, y⟩ = x₁y₁ + x₂y₂ + ... + xₙyₙ$
• C^n with complex inner product: $⟨x, y⟩ = x₁y₁* + x₂y₂* + ... + xₙyₙ*$
• Function spaces with integral-based inner products
  • Continuous functions on [a,b]: $⟨f, g⟩ = ∫[a,b] f(x)g(x)dx$
• Polynomial spaces with various inner products
  • Legendre polynomials: $⟨P, Q⟩ = ∫[-1,1] P(x)Q(x)dx$
• Matrix spaces with inner products
  • Frobenius inner product: $⟨A, B⟩ = tr(A*B)$ for complex matrices

Construction and Verification

• Constructing new spaces from existing ones
  • Direct sums of inner product spaces
  • Tensor products of inner product spaces
• Verifying proposed function satisfies inner product properties
  • Conjugate symmetry
  • Linearity in first argument
  • Positive definiteness
• Completion concept for constructing Hilbert spaces from pre-Hilbert spaces
  • Example: Completing rational numbers to get real numbers

Inner Products for Geometry

Angles and Projections

• Calculate angles between vectors: $cos θ = ⟨x, y⟩ / (||x|| ||y||)$
  • Example: Angle between (1, 1) and (1, -1) in R^2
• Compute orthogonal projections: $proj_u(v) = (⟨v, u⟩ / ⟨u, u⟩) u$
  • Example: Projection of (3, 4) onto (1, 1) in R^2
• Gram-Schmidt process constructs orthonormal basis from linearly independent set
  • Example: Orthonormalizing {(1, 1, 0), (1, 0, 1), (0, 1, 1)} in R^3

Geometric Applications

• Calculate distance between vectors: $d(x, y) = ||x - y|| = √⟨x - y, x - y⟩$
• Define and compute volume of parallelepipeds via Gram determinant
• Analyze orthogonal complement of subspace using inner products
  • Example: Orthogonal complement of xy-plane in R^3
• Solve least squares problems and compute best approximations
  • Example: Finding best-fit line for set of data points
• Applications in quantum mechanics
  • Inner products used to calculate expectation values of observables