Inner product spaces let us measure angles and distances between vectors. This section dives into orthogonal complements - subspaces containing vectors perpendicular to a given subspace. We'll see how these complements relate to the original space's structure.

We'll also explore orthogonal projections, which find the closest vector in a subspace to a given vector. This concept is crucial for solving least-squares problems and decomposing vectors into orthogonal components.

Orthogonal Complements and Properties

Definition and Basic Properties

• Orthogonal complement of subspace W in inner product space V denoted W⊥ contains all vectors in V orthogonal to every vector in W
• For finite-dimensional inner product space V and subspace W, (W⊥)⊥ = W
• Dimension relationship dim(W) + dim(W⊥) = dim(V) holds for finite-dimensional inner product spaces
• Orthogonal complement of a subspace forms a subspace of the inner product space
• Zero subspace {0} has orthogonal complement V, and V has orthogonal complement {0}

Properties Involving Multiple Subspaces

• For subspaces U and W in V, (U + W)⊥ = U⊥ ∩ W⊥
• Intersection and sum relationship (U ∩ W)⊥ = U⊥ + W⊥ holds for subspaces U and W
• Linear transformation T : V → W between inner product spaces yields ker(T) = (range(T))⊥
• Adjoint T* of linear transformation T satisfies range(T*) = (ker(T))⊥

Orthogonal Projections onto Subspaces

Computation and Properties

• Orthogonal projection of vector v onto subspace W represents closest vector in W to v
• Formula for orthogonal projection onto W with orthonormal basis {u₁, ..., uₖ} $proj_W(v) = \sum_{i=1}^k \langle v, u_i \rangle u_i$
• Projection matrix P for orthogonal projection onto W $P = U(U^*U)^{-1}U^*$ where U forms basis for W
• Orthogonal projection P onto W satisfies P² = P (idempotent) and P = P (self-adjoint)
• Orthogonal projection onto W⊥ calculated as v - proj_W(v)
• Error vector e = v - proj_W(v) orthogonal to all vectors in W

Orthogonal Decomposition

• Vector v in V decomposes as v = proj_W(v) + proj_W⊥(v)
• Decomposition represents v as sum of components in W and W⊥
• Orthogonal decomposition unique for each vector v in V

Orthogonal Decomposition Theorem

Statement and Proof

• Theorem states every vector v in V uniquely expressed as v = w + w⊥, w in W and w⊥ in W⊥
• Existence proof constructs w = proj_W(v) and w⊥ = v - proj_W(v)
• Uniqueness proof assumes two decompositions v = w₁ + w₁⊥ = w₂ + w₂⊥ and shows w₁ = w₂ and w₁⊥ = w₂⊥
• Proof utilizes inner product properties and orthogonal complement definition

Implications and Applications

• Theorem implies V = W ⊕ W⊥ (direct sum) for any subspace W of V
• Unique decomposition of vectors into projections onto W and W⊥
• Fundamental theorem in understanding inner product space structure
• Applications in linear algebra, functional analysis, and quantum mechanics (state vector decomposition)

Least-Squares Problems with Projections

Problem Formulation and Solution

• Least-squares problem minimizes ||Ax - b||² for matrix A and vector b
• Normal equations AAx = Ab characterize least-squares solution
• Solution x̂ given by $\hat{x} = (A^*A)^{-1}A^*b$ when AA invertible
• Geometrically, solution finds closest vector Ax̂ in column space of A to b
• Orthogonal projection of b onto column space of A $proj_{col(A)}(b) = A(A^*A)^{-1}A^*b$
• Residual vector r = b - Ax̂ orthogonal to column space of A

Applications and Significance

• Data fitting applications (linear regression, polynomial fitting)
• Model matrix A represents predictor variables, b represents observed data
• Signal processing uses (noise reduction, signal approximation)
• Parameter estimation in scientific and engineering fields (system identification)
• Statistical analysis applications (ANOVA, multiple regression)