Questions and Answers
What is the standard definition of an inner product?
Hermitian inner product ⟨u, v⟩ = v ∗ u.conjugate transpose of v multiplied by u.
Give four identities that hold under the conjugate transpose.
(i) (A∗) ∗ = A. (ii) (λA) ∗ = λ (conjugate)(A∗). (iii) (A + B) ∗ = A∗ + B∗. (iv) (AB) ∗ = B ∗A ∗.
Give four identities that hold under the inner product.
(a) ⟨u + v, w⟩ = ⟨u, w⟩ + ⟨v, w⟩. (b) ⟨αu, v⟩ = α ⟨u, v⟩. (c) ⟨u, v⟩ = (⟨v, u⟩ conjugate). (d) ⟨u, u⟩ ⩾ 0, and ⟨u, u⟩ = 0 ⇔ u = 0.
What do we call a vector space equipped with an inner product?
Signup and view all the answers
Give four examples of inner product spaces.
Signup and view all the answers
What is a property about subspaces and inner product spaces?
Signup and view all the answers
Give a fundamental fact about how matrices interact with the Hermitian inner product.
Signup and view all the answers
What makes two vectors orthogonal?
Signup and view all the answers
What is the norm of a vector?
Signup and view all the answers
What is the definition of a unit vector?
Signup and view all the answers
What makes a set of an inner product space orthogonal?
Signup and view all the answers
What makes a set of an inner product space orthonormal?
Signup and view all the answers
Give three properties of an inner product space in regards to norms.
Signup and view all the answers
What is the definition of an orthonormal basis?
Signup and view all the answers
What is a property of orthogonal sets?
Signup and view all the answers
What do we use to turn a basis into an orthonormal basis?
Signup and view all the answers
What is the definition of an isometry?
Signup and view all the answers
What are three statements that are equivalent to describing a linear transformation as an isometry?
Signup and view all the answers
What is a useful specific example of an isometry?
Signup and view all the answers
What is the easiest way to verify that a transformation is an isometry?
Signup and view all the answers
What is a unitary matrix?
Signup and view all the answers
What is the name for a unitary matrix over R?
Signup and view all the answers
Study Notes
Inner Product Definition
- The Hermitian inner product is defined as ⟨u, v⟩ = v ∗ u.conjugate transpose multiplied by u.
Conjugate Transpose Identities
- Identities include:
- (A∗) ∗ = A
- (λA) ∗ = λ (conjugate)(A∗)
- (A + B) ∗ = A∗ + B∗
- (AB) ∗ = B ∗A ∗
Inner Product Identities
- Key properties of inner products:
- ⟨u + v, w⟩ = ⟨u, w⟩ + ⟨v, w⟩
- ⟨αu, v⟩ = α ⟨u, v⟩
- ⟨u, v⟩ = (⟨v, u⟩ conjugate)
- ⟨u, u⟩ ⩾ 0, with equality to 0 if and only if u = 0
Inner Product Space
- An inner product space is a vector space with an inner product satisfying:
- Additivity: ⟨u + v, w⟩ = ⟨u, w⟩ + ⟨v, w⟩
- Scalar multiplication: ⟨αu, v⟩ = α ⟨u, v⟩
- Conjugate symmetry: ⟨u, v⟩ = (⟨v, u⟩ conjugate)
- Non-negativity: ⟨u, u⟩ ⩾ 0, with zero norm indicating the zero vector.
Examples of Inner Product Spaces
- Rn over R with the dot product
- Cn over C with the Hermitian inner product
- Pn over R, using integrals for inner products
- Mmn over F with the Frobenius inner product
Subspaces of Inner Product Spaces
- Any subspace of an inner product space can itself be considered an inner product space.
Interaction with Hermitian Inner Product
- For matrices A and vectors u, v in F^n, the transfer relation is ⟨Au, v⟩ = ⟨u, A∗ v⟩.
Orthogonality
- Two vectors are orthogonal if their inner product is zero: ⟨u, v⟩ = 0.
Norm of a Vector
- The norm is defined as the square root of the inner product of a vector with itself: ||u|| = √(⟨u, u⟩).
Unit Vector Definition
- A unit vector has a norm of 1.
Orthogonal Sets
- A set S is orthogonal if for any distinct u, v in S, the inner product ⟨u, v⟩ = 0.
Orthonormal Sets
- A set is orthonormal if it is both orthogonal and consists solely of unit vectors.
Norm Properties
- Norm properties include:
- Scaling: ||λv|| = |λ| ||v||
- Normalization: For non-zero v, (||v||−1)v is a unit vector.
- Orthogonality preservation: For an orthogonal set S with non-zero vectors, the set of normalized vectors {||v||−1 v | v ∈ S} is orthonormal.
Orthonormal Basis Definition
- An orthonormal basis consists of orthogonal unit vectors spanning the vector space V.
Linear Independence of Orthogonal Sets
- Any orthogonal set of non-zero vectors is linearly independent.
Gram-Schmidt Process
- This process is used to convert a basis into an orthonormal basis.
Isometry Definition
- An isometry is a surjective linear transformation T: V → W where ||T(v)|| = ||v|| for all v in V.
Isometry Characterizations
- Equivalent statements for isometries:
- T preserves lengths: ||T(v)|| = ||v||.
- T is a surjective isomorphism preserving inner products: ⟨T(v1), T(v2)⟩ = ⟨v1, v2⟩ for all v1, v2 in V.
- T transforms an orthonormal basis of V into an orthonormal basis of W.
Example of Isometry
- Mapping a vector to its coordinate vector in an orthonormal basis is a concrete example of an isometry.
Verifying Isometry
- To check if a transformation is an isometry, confirm it maps an orthonormal basis of the domain to an orthonormal basis of the codomain.
Unitary Matrix Definition
- A unitary matrix has orthonormal columns in Fn.
Orthogonal Matrix Definition
- A unitary matrix over R is referred to as an orthogonal matrix.
Equivalence of Unitary Matrices
- Equivalent statements for a matrix being unitary include properties related to preserving inner products and norms.
Studying That Suits You
Use AI to generate personalized quizzes and flashcards to suit your learning preferences.
Description
Test your knowledge on inner product spaces with these flashcards. Dive into definitions, properties, and identities related to Hermitian inner products and conjugate transposes. Perfect for students studying linear algebra concepts.