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
Top images from around the web for Definition and Key Properties Representation of dot product of a vector - Mathematics Stack Exchange View original
Is this image relevant?
1 of 3
Top images from around the web for Definition and Key Properties Representation of dot product of a vector - Mathematics Stack Exchange View original
Is this image relevant?
1 of 3
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 ⟩ ⟨x, y⟩ = ⟨y, x⟩ ⟨ x , y ⟩ = ⟨ y , x ⟩ for all x, y ∈ V
Linearity in first argument means ⟨ α x + β y , z ⟩ = α ⟨ x , z ⟩ + β ⟨ y , z ⟩ ⟨αx + βy, z⟩ = α⟨x, z⟩ + β⟨y, z⟩ ⟨ αx + β y , z ⟩ = α ⟨ x , z ⟩ + β ⟨ y , z ⟩ for all x, y, z ∈ V and α, β ∈ F
Positive definiteness ensures ⟨ x , x ⟩ ≥ 0 ⟨x, x⟩ ≥ 0 ⟨ 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 ⟩ ||x|| = √⟨x, x⟩ ∣∣ 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 ⟨x, y⟩ = 0 ⟨ 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 ⟩ |⟨x, y⟩|² ≤ ⟨x, x⟩⟨y, y⟩ ∣ ⟨ x , y ⟩ ∣ 2 ≤ ⟨ x , x ⟩ ⟨ y , y ⟩ for all x, y in inner product space
Proof typically involves quadratic function f ( t ) = ⟨ x + t y , x + t y ⟩ f(t) = ⟨x + ty, x + ty⟩ f ( t ) = ⟨ x + t y , x + t y ⟩
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 ∣ ∣ ||x + y|| ≤ ||x|| + ||y|| ∣∣ 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: c o s θ = ⟨ x , y ⟩ / ( ∣ ∣ x ∣ ∣ ∣ ∣ y ∣ ∣ ) cos θ = ⟨x, y⟩ / (||x|| ||y||) 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 ⟩ = x 1 y 1 + x 2 y 2 + . . . + x n y n ⟨x, y⟩ = x₁y₁ + x₂y₂ + ... + xₙyₙ ⟨ x , y ⟩ = x 1 y 1 + x 2 y 2 + ... + x n y n
C^n with complex inner product: ⟨ x , y ⟩ = x 1 y 1 ∗ + x 2 y 2 ∗ + . . . + x n y n ∗ ⟨x, y⟩ = x₁y₁* + x₂y₂* + ... + xₙyₙ* ⟨ x , y ⟩ = x 1 y 1 ∗ + x 2 y 2 ∗ + ... + x n y n ∗
Function spaces with integral-based inner products
Continuous functions on [a,b]: ⟨ f , g ⟩ = ∫ [ a , b ] f ( x ) g ( x ) d x ⟨f, g⟩ = ∫[a,b] f(x)g(x)dx ⟨ f , g ⟩ = ∫ [ a , b ] f ( x ) g ( x ) d x
Polynomial spaces with various inner products
Legendre polynomials: ⟨ P , Q ⟩ = ∫ [ − 1 , 1 ] P ( x ) Q ( x ) d x ⟨P, Q⟩ = ∫[-1,1] P(x)Q(x)dx ⟨ P , Q ⟩ = ∫ [ − 1 , 1 ] P ( x ) Q ( x ) d x
Matrix spaces with inner products
Frobenius inner product: ⟨ A , B ⟩ = t r ( A ∗ B ) ⟨A, B⟩ = tr(A*B) ⟨ A , B ⟩ = t r ( 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: c o s θ = ⟨ x , y ⟩ / ( ∣ ∣ x ∣ ∣ ∣ ∣ y ∣ ∣ ) cos θ = ⟨x, y⟩ / (||x|| ||y||) cos θ = ⟨ x , y ⟩ / ( ∣∣ x ∣∣∣∣ y ∣∣ )
Example: Angle between (1, 1) and (1, -1) in R^2
Compute orthogonal projections: p r o j u ( v ) = ( ⟨ v , u ⟩ / ⟨ u , u ⟩ ) u proj_u(v) = (⟨v, u⟩ / ⟨u, u⟩) u p ro j 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 ⟩ d(x, y) = ||x - y|| = √⟨x - y, x - y⟩ 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