➗Linear Algebra and Differential Equations Unit 6 – Inner Products and Orthogonality

Key Concepts

• Inner products generalize the notion of the dot product to abstract vector spaces
• Orthogonality refers to the relationship between vectors that are perpendicular to each other
• Inner products allow for the computation of lengths, distances, and angles between vectors in a vector space
• Orthogonal projections decompose a vector into components that are parallel and perpendicular to a given subspace
• The Gram-Schmidt process constructs an orthonormal basis from a linearly independent set of vectors
• Inner products and orthogonality have applications in various areas of linear algebra, such as least squares approximation and eigenvalue problems
• Orthogonality plays a crucial role in the study of differential equations, particularly in the context of Sturm-Liouville theory and Fourier series

Inner Product Basics

• An inner product is a function that assigns a scalar value to a pair of vectors in a vector space
• Denoted as $\langle \mathbf{u}, \mathbf{v} \rangle$, where $\mathbf{u}$ and $\mathbf{v}$ are vectors
• For real vector spaces, the inner product is often defined as the dot product: $\langle \mathbf{u}, \mathbf{v} \rangle = \mathbf{u} \cdot \mathbf{v} = u_1v_1 + u_2v_2 + \ldots + u_nv_n$
• The inner product of a vector with itself gives the square of its length: $\langle \mathbf{u}, \mathbf{u} \rangle = \|\mathbf{u}\|^2$
• The inner product can be used to calculate the angle $\theta$ between two vectors using the formula: $\cos \theta = \frac{\langle \mathbf{u}, \mathbf{v} \rangle}{\|\mathbf{u}\| \|\mathbf{v}\|}$
• Inner products can be defined for complex vector spaces, with the conjugate of the first vector used in the computation

Properties of Inner Products

• Symmetry: $\langle \mathbf{u}, \mathbf{v} \rangle = \langle \mathbf{v}, \mathbf{u} \rangle$ for real vector spaces, and $\langle \mathbf{u}, \mathbf{v} \rangle = \overline{\langle \mathbf{v}, \mathbf{u} \rangle}$ for complex vector spaces
• Linearity in the second argument:
  • $\langle \mathbf{u}, \mathbf{v} + \mathbf{w} \rangle = \langle \mathbf{u}, \mathbf{v} \rangle + \langle \mathbf{u}, \mathbf{w} \rangle$
  • $\langle \mathbf{u}, c\mathbf{v} \rangle = c\langle \mathbf{u}, \mathbf{v} \rangle$, where $c$ is a scalar
• Positive definiteness: $\langle \mathbf{u}, \mathbf{u} \rangle \geq 0$, with equality if and only if $\mathbf{u} = \mathbf{0}$
• Cauchy-Schwarz inequality: $|\langle \mathbf{u}, \mathbf{v} \rangle| \leq \|\mathbf{u}\| \|\mathbf{v}\|$, with equality if and only if $\mathbf{u}$ and $\mathbf{v}$ are linearly dependent
• Triangle inequality: $\|\mathbf{u} + \mathbf{v}\| \leq \|\mathbf{u}\| + \|\mathbf{v}\|$

Orthogonality Explained

• Two vectors $\mathbf{u}$ and $\mathbf{v}$ are orthogonal (perpendicular) if their inner product is zero: $\langle \mathbf{u}, \mathbf{v} \rangle = 0$
• A set of vectors $\{\mathbf{v}_1, \mathbf{v}_2, \ldots, \mathbf{v}_n\}$ is called an orthogonal set if every pair of distinct vectors in the set is orthogonal
• An orthogonal set of vectors is linearly independent
• If an orthogonal set of vectors has unit length (i.e., $\|\mathbf{v}_i\| = 1$ for all $i$), it is called an orthonormal set
• Orthogonal matrices are square matrices whose columns (or rows) form an orthonormal set
  • Orthogonal matrices have the property that their inverse is equal to their transpose: $A^{-1} = A^T$
• Orthogonal complements: For a subspace $W$ of a vector space $V$, the orthogonal complement $W^\perp$ is the set of all vectors in $V$ that are orthogonal to every vector in $W$

Orthogonal Projections

• An orthogonal projection of a vector $\mathbf{u}$ onto a subspace $W$ is the closest point in $W$ to $\mathbf{u}$
• The orthogonal projection of $\mathbf{u}$ onto $W$ is the unique vector $\mathbf{w} \in W$ such that $\mathbf{u} - \mathbf{w}$ is orthogonal to every vector in $W$
• The projection matrix $P$ onto a subspace spanned by an orthonormal basis $\{\mathbf{v}_1, \mathbf{v}_2, \ldots, \mathbf{v}_k\}$ is given by $P = \mathbf{v}_1\mathbf{v}_1^T + \mathbf{v}_2\mathbf{v}_2^T + \ldots + \mathbf{v}_k\mathbf{v}_k^T$
• The orthogonal projection of $\mathbf{u}$ onto $W$ can be computed as $\text{proj}_W(\mathbf{u}) = P\mathbf{u}$
• Orthogonal projections have applications in least squares approximation, signal processing, and data compression

Gram-Schmidt Process

• The Gram-Schmidt process is an algorithm for constructing an orthonormal basis from a linearly independent set of vectors
• Given a linearly independent set $\{\mathbf{u}_1, \mathbf{u}_2, \ldots, \mathbf{u}_n\}$, the Gram-Schmidt process produces an orthonormal set $\{\mathbf{v}_1, \mathbf{v}_2, \ldots, \mathbf{v}_n\}$ spanning the same subspace
• The process works by sequentially orthogonalizing each vector with respect to the previous orthonormal vectors and then normalizing the result
• The orthogonalization step for the $k$-th vector is given by $\mathbf{u}_k' = \mathbf{u}_k - \sum_{i=1}^{k-1} \langle \mathbf{u}_k, \mathbf{v}_i \rangle \mathbf{v}_i$
• The normalization step is given by $\mathbf{v}_k = \frac{\mathbf{u}_k'}{\|\mathbf{u}_k'\|}$
• The Gram-Schmidt process is numerically unstable for nearly linearly dependent input vectors, and the modified Gram-Schmidt process is often used instead

Applications in Linear Algebra

• Inner products and orthogonality are fundamental concepts in linear algebra with numerous applications
• Least squares approximation: Finding the best approximation of a vector in a subspace using orthogonal projections
• Principal component analysis (PCA): Identifying the orthogonal directions of maximum variance in a dataset for dimensionality reduction
• Singular value decomposition (SVD): Factorizing a matrix into orthogonal matrices and a diagonal matrix of singular values, used in data compression and noise reduction
• Eigenvalue problems: Orthogonality of eigenvectors corresponding to distinct eigenvalues of a symmetric matrix
• Orthogonal diagonalization: Diagonalizing a symmetric matrix using an orthogonal matrix of eigenvectors

Connections to Differential Equations

• Orthogonality plays a crucial role in the study of differential equations, particularly in the context of Sturm-Liouville theory and Fourier series
• Sturm-Liouville theory deals with eigenvalue problems for certain types of linear differential equations
  • The eigenfunctions of a Sturm-Liouville problem form an orthogonal basis for the function space
• Fourier series represent functions as infinite sums of orthogonal trigonometric functions (sines and cosines)
  • The coefficients of a Fourier series can be computed using inner products of the function with the basis functions
• Orthogonal polynomials (e.g., Legendre polynomials, Chebyshev polynomials) are solutions to certain Sturm-Liouville problems and are used in approximation theory and numerical analysis
• The wave equation and the heat equation can be solved using Fourier series or other orthogonal function expansions
• Orthogonality is also important in the study of partial differential equations, such as in the method of separation of variables and the finite element method