Inner Product Spaces, Gram-Schmidt, Projections and Adjoints

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1 · Prerequisites

Abstract Algebra

Combinatorics

Finite Counting, Factorials and Binomial Coefficients

Coxeter Groups

Symmetric Groups, Cycle Decomposition and the Sign Homomorphism

Linear Algebra

Real Analysis

Set Theory

2 · Logical flowchart

3 · Items

Definition

Every invertible real or complex square matrix has a unique factorisation A=QR with Q orthogonal or unitary and R upper triangular with positive real diagonal

Theorem

For a linear map T:V→ W between finite-dimensional inner-product spaces, x minimises ‖ Tx-b‖ if and only if T^(Tx-b)=0, equivalently T^Tx=T^*b; minimisers exist and any two differ by an element of ker T

Lemma

For an endomorphism in finite dimension, preserving lengths, preserving inner products, carrying orthonormal bases to orthonormal bases, and T^*T=I are equivalent

Corollary

Orthogonal and unitary operators form groups, and their determinants have modulus one

Sources

Inner Product Spaces, Gram-Schmidt, Projections and Adjoints

3 · Items

Real and complex inner product spaces, with the inner product linear in the first argument

The norm ‖ v‖=√⟨ v,v⟩ induced by a real or complex inner product

The standard formulas ⟨ x,y⟩=∑_k<nxₖ yₖ on ℝⁿ and ∑_k<nxₖyₖ on ℂⁿ are inner products

Inner products separate vectors, and the induced norm is homogeneous: ‖λ v‖=|λ|‖ v‖

Cauchy–Schwarz: |⟨ u,v⟩|≤‖ u‖‖ v‖, with equality exactly for linearly dependent vectors

The inner-product norm is definite, homogeneous, and satisfies the triangle inequality

Pythagoras, the parallelogram identity, and the real and complex polarisation identities

On ℝⁿ, abstract Cauchy–Schwarz is exactly the published finite-sum Cauchy–Schwarz inequality

Orthogonal vectors and subspaces, orthogonal and orthonormal sets, and orthonormal bases

The orthogonal complement W^⊥={v:⟨ v,w⟩=0 for all w∈ W}

Every finite orthogonal list of nonzero vectors is linearly independent

Gram–Schmidt turns every finite independent list into an orthonormal list with the same successive spans

Every finite-dimensional real or complex inner product space has an orthonormal basis

Bessel's inequality for a finite orthonormal list and Parseval's identity for an orthonormal basis

For a subspace W of a finite-dimensional inner product space, V=W⊕ W^⊥

In finite dimension, W^⊥⊥=W and dim W+dim W^⊥=dim V

The orthogonal projection P_Wv is the W-component in V=W⊕ W^⊥

Orthogonal projection is linear, and an orthonormal basis (eᵢ) of W gives P_Wv=∑ᵢ⟨ v,eᵢ⟩ eᵢ

The orthogonal projection is the unique nearest point in the subspace

The Gram matrix G(v₀,…,vᵣ₋₁)=(⟨ vᵢ,vⱼ⟩)_i,j<r and Gram determinant, with empty value 1

A Gram determinant is nonnegative and is positive exactly when the vector list is linearly independent

Every invertible real or complex square matrix has a unique factorisation A=QR with Q orthogonal or unitary and R upper triangular with positive real diagonal

Finite-dimensional Riesz representation: every functional is uniquely v↦⟨ v,w⟩

The adjoint T^:W→ V is characterised by ⟨ Tv,w⟩_W=⟨ v,T^w⟩_V

Every linear map between finite-dimensional inner product spaces has a unique adjoint

Adjoints satisfy (S+T)^=S^+T^, (λ T)^=λ̄ T^, (ST)^=T^S^, and T^**=T

In orthonormal bases, the matrix of the adjoint is the conjugate transpose of the matrix

ker T^=(imT)^⊥ and imT^=(ker T)^⊥ in finite dimension

For a linear map T:V→ W between finite-dimensional inner-product spaces, x minimises ‖ Tx-b‖ if and only if T^(Tx-b)=0, equivalently T^Tx=T^*b; minimisers exist and any two differ by an element of ker T

If W is T-invariant, then W^⊥ is T^*-invariant

An endomorphism is an orthogonal projection exactly when it is idempotent and self-adjoint

Linear isometries, and orthogonal or unitary operators on finite-dimensional inner product spaces

For an endomorphism in finite dimension, preserving lengths, preserving inner products, carrying orthonormal bases to orthonormal bases, and T^*T=I are equivalent

Orthogonal and unitary operators form groups, and their determinants have modulus one

Inner Product Spaces, Gram-Schmidt, Projections and Adjoints

3 · Items

Real and complex inner product spaces, with the inner product linear in the first argument

The norm ‖ v‖=√⟨ v,v⟩ induced by a real or complex inner product

The standard formulas ⟨ x,y⟩=∑_k<nxₖ yₖ on ℝⁿ and ∑_k<nxₖyₖ on ℂⁿ are inner products

Inner products separate vectors, and the induced norm is homogeneous: ‖λ v‖=|λ|‖ v‖

Cauchy–Schwarz: |⟨ u,v⟩|≤‖ u‖‖ v‖, with equality exactly for linearly dependent vectors

The inner-product norm is definite, homogeneous, and satisfies the triangle inequality

Pythagoras, the parallelogram identity, and the real and complex polarisation identities

On ℝⁿ, abstract Cauchy–Schwarz is exactly the published finite-sum Cauchy–Schwarz inequality

Orthogonal vectors and subspaces, orthogonal and orthonormal sets, and orthonormal bases

The orthogonal complement W^⊥={v:⟨ v,w⟩=0 for all w∈ W}

Every finite orthogonal list of nonzero vectors is linearly independent

Gram–Schmidt turns every finite independent list into an orthonormal list with the same successive spans

Every finite-dimensional real or complex inner product space has an orthonormal basis

Bessel's inequality for a finite orthonormal list and Parseval's identity for an orthonormal basis

For a subspace W of a finite-dimensional inner product space, V=W⊕ W^⊥

In finite dimension, W^⊥⊥=W and dim W+dim W^⊥=dim V

The orthogonal projection P_Wv is the W-component in V=W⊕ W^⊥

Orthogonal projection is linear, and an orthonormal basis (eᵢ) of W gives P_Wv=∑ᵢ⟨ v,eᵢ⟩ eᵢ

The orthogonal projection is the unique nearest point in the subspace

The Gram matrix G(v₀,…,vᵣ₋₁)=(⟨ vᵢ,vⱼ⟩)_i,j<r and Gram determinant, with empty value 1

A Gram determinant is nonnegative and is positive exactly when the vector list is linearly independent

Every invertible real or complex square matrix has a unique factorisation A=QR with Q orthogonal or unitary and R upper triangular with positive real diagonal

Finite-dimensional Riesz representation: every functional is uniquely v↦⟨ v,w⟩

The adjoint T^*:W→ V is characterised by ⟨ Tv,w⟩_W=⟨ v,T^*w⟩_V

Every linear map between finite-dimensional inner product spaces has a unique adjoint

Adjoints satisfy (S+T)^*=S^*+T^*, (λ T)^*=λ̄ T^*, (ST)^*=T^*S^*, and T^**=T

In orthonormal bases, the matrix of the adjoint is the conjugate transpose of the matrix

ker T^*=(imT)^⊥ and imT^*=(ker T)^⊥ in finite dimension

For a linear map T:V→ W between finite-dimensional inner-product spaces, x minimises ‖ Tx-b‖ if and only if T^*(Tx-b)=0, equivalently T^*Tx=T^*b; minimisers exist and any two differ by an element of ker T

If W is T-invariant, then W^⊥ is T^*-invariant

An endomorphism is an orthogonal projection exactly when it is idempotent and self-adjoint

Linear isometries, and orthogonal or unitary operators on finite-dimensional inner product spaces

For an endomorphism in finite dimension, preserving lengths, preserving inner products, carrying orthonormal bases to orthonormal bases, and T^*T=I are equivalent

Orthogonal and unitary operators form groups, and their determinants have modulus one

The adjoint T^:W→ V is characterised by ⟨ Tv,w⟩_W=⟨ v,T^w⟩_V

Adjoints satisfy (S+T)^=S^+T^, (λ T)^=λ̄ T^, (ST)^=T^S^, and T^**=T

ker T^=(imT)^⊥ and imT^=(ker T)^⊥ in finite dimension

For a linear map T:V→ W between finite-dimensional inner-product spaces, x minimises ‖ Tx-b‖ if and only if T^(Tx-b)=0, equivalently T^Tx=T^*b; minimisers exist and any two differ by an element of ker T