Linear Algebra: Inner Product Spaces Flashcards

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?

An inner product space. Signup and view all the answers

Give four examples of inner product spaces.

Rn over R with the usual inner product (dot product), Cn over C with the Hermitian inner product (conjugate transpose), Pn over R with inner product defined as integral of p*q, Mmn over F with the Frobenius inner product. Signup and view all the answers

What is a property about subspaces and inner product spaces?

Any subspace of an inner product space can be considered to be an inner product space. Signup and view all the answers

Give a fundamental fact about how matrices interact with the Hermitian inner product.

For any A ∈ Mn(F), u ∈ Fn and v ∈ Fn, ⟨Au, v⟩ = ⟨u, A∗v⟩. Signup and view all the answers

What makes two vectors orthogonal?

The inner product is 0. Signup and view all the answers

What is the norm of a vector?

Square root of the inner product of it with itself, sq root(⟨u, u⟩). Signup and view all the answers

What is the definition of a unit vector?

Norm is 1. Signup and view all the answers

What makes a set of an inner product space orthogonal?

For all u, v ∈ S, u ≠ v ⇒ ⟨u, v⟩ = 0. Signup and view all the answers

What makes a set of an inner product space orthonormal?

A set of vectors in V is orthonormal if it is orthogonal and consists only of unit vectors. Signup and view all the answers

Give three properties of an inner product space in regards to norms.

(a) For any vector v ∈ V and any scalar λ ∈ F, ∥λv∥ = |λ| ∥v∥. (b) If v ∈ V is non-zero then ∥v∥ −1 v is a unit vector. (c) If S ⊆ Rn is orthogonal and 0 ≠ S, then {∥v∥ −1 v|v ∈ S} is an orthonormal set. Signup and view all the answers

What is the definition of an orthonormal basis?

An orthonormal basis of V is a basis of V which is an orthonormal set. Signup and view all the answers

What is a property of orthogonal sets?

Any orthogonal set S of non-zero vectors is linearly independent. Signup and view all the answers

What do we use to turn a basis into an orthonormal basis?

The Gram-Schmidt process. Signup and view all the answers

What is the definition of an isometry?

A linear transformation T: V → W is an isometry if T is surjective and ∥T(v)∥ = ∥v∥. Signup and view all the answers

What are three statements that are equivalent to describing a linear transformation as an isometry?

T is an isomorphism such that ∀v ∈ V, ∥T(v)∥ = ∥v∥. T is surjective and ∀v1, v2 ∈ V, ⟨T(v1), T(v2)⟩ = ⟨v1, v2⟩. If dim V = n and {v1, v2,..., vn} is an orthonormal basis of V, these conditions are also equivalent to: {T(v1), T(v2),..., T(vn)} is an orthonormal basis of W. Signup and view all the answers

What is a useful specific example of an isometry?

Mapping a vector onto its coordinate vector in an orthonormal basis is always an isometry. Signup and view all the answers

What is the easiest way to verify that a transformation is an isometry?

Transform an orthonormal basis for the domain and verify it is an orthonormal basis for the codomain. Signup and view all the answers

What is a unitary matrix?

A square matrix where columns form an orthonormal basis of Fn. Signup and view all the answers

What is the name for a unitary matrix over R?

An orthogonal matrix. 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.

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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.