Alphabeta MathInner Product Spaces and Orthogonality: Examples and Counterexamples
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✓ 16 results · all verified · 10 also independently AI-judged
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1 · Prerequisites
Abstract Algebra
- Binary Operations, Monoids, Groups and Subgroups
- Cosets, Index and Lagrange's Theorem
- Divisibility, Euclidean Domains, Principal Ideal Domains and Unique Factorisation
- Group Actions, Orbits, Stabilisers and Cayley's Theorem
- Ideals, Quotient Rings and the Isomorphism Theorems for Rings
- Normal Subgroups and Quotient Groups
- Polynomial Rings, the Division Algorithm and Roots
- Rings, Subrings, Integral Domains and Fields
- Simple Field Extensions and the Construction of the Complex Numbers
Linear Algebra
- Determinants of Matrices over a Commutative Ring
- Dual Spaces, Bilinear and Quadratic Forms, and Sylvester's Law of Inertia
- Inner Product Spaces, Gram-Schmidt, Projections and Adjoints
- Linear Independence, Bases and Dimension
- Linear Transformations, Rank-Nullity and Quotient Spaces
- Matrices, the Matrix of a Linear Map, and Change of Basis
- The Determinant of a Linear Operator, Cofactors and Cramer's Rule
- Vector Spaces, Linear Subspaces, Span and Direct Sums
Real Analysis
2 · Logical flowchart
3 · Items
Example
Positive coordinate weights define an inner product and change lengths and projections
Example
The Frobenius inner product on a real or complex matrix space
Example
Evaluation at four distinct real points gives an inner product on polynomials of degree at most three
Example
Gram–Schmidt on an explicit basis of ℝ³
Example
Gram–Schmidt on an explicit basis of ℂ² with conjugation visible
Example
A worked QR factorisation of an invertible real matrix
Example
Bessel's inequality is strict for a vector outside the span of a proper orthonormal set
Example
Projection onto a plane in ℝ³ and the nearest-point calculation
Example
The Riesz representative of an explicit functional on ℂ³
Example
The adjoint of an explicit map ℂ²→ℂ³ is its conjugate-transpose matrix
Example
The Gram determinant of two vectors is ‖ u‖²‖ v‖²-|⟨ u,v⟩|² and detects dependence
Counterexample
A nondegenerate indefinite symmetric form can have W∩ W^⊥≠0
False statement
FALSE: Every complex inner product satisfies ⟨ u,v⟩=⟨ v,u⟩
False statement
FALSE: Every orthogonal set is an orthonormal basis
False statement
FALSE: Every idempotent endomorphism of an inner product space is an orthogonal projection
False statement
FALSE: On a complex inner product space, (λ T)^*=λ T^* for every scalar
Sources
- Sergei Treil, Linear Algebra Done Wrong, Example 5.1.3
- Sergei Treil, Linear Algebra Done Wrong, §5.3.2
- Sheldon Axler, Linear Algebra Done Right, 4th ed., definition 6.2
- Sheldon Axler, Linear Algebra Done Right, 4th ed., definitions 6.22 and 6.27
- Sheldon Axler, Linear Algebra Done Right, 4th ed., §§6C and 7A
- Sheldon Axler, Linear Algebra Done Right, 4th ed., result 7.5