Vector Space Notation

Basic notation for vector spaces and their elements:

1. Common Vector Spaces

   Standard vector spaces over real numbers, complex numbers, and general fields.

2. Vector Space Membership

   Indicating that a vector belongs to a vector space, and a vector space is a subset of another.

3. Function Spaces

   Common function spaces: continuous functions, square-integrable functions, and Sobolev spaces.

Basis and Dimension

Representing basis vectors and dimensions:

1. Standard Basis

   The standard basis for ℝⁿ.

2. General Basis

   A general basis for an n-dimensional vector space.

3. Dimension

   The dimension of a vector space V.

Linear Combinations and Span

1. Linear Combination

   A linear combination of vectors.

2. Span

   The span of a set of vectors.

3. Linear Independence

   The definition of linear independence.

Subspaces and Direct Sums

1. Subspace

   U is a subspace of V.

2. Direct Sum

   V is the direct sum of subspaces U and W.

3. Null Space and Range

Linear Transformations

1. Linear Transformation Definition

   A linear transformation from vector space V to W.

2. Matrix Representation

   Matrix representation of a linear transformation T with respect to bases ℬ and ℭ.

3. Composition of Linear Transformations

Eigenvalues and Eigenvectors

1. Eigenvalue Equation

   The defining equation for eigenvalue λ and eigenvector v.

2. Characteristic Polynomial

3. Eigenspace

   The eigenspace corresponding to eigenvalue λ.

Inner Product Spaces

1. Inner Product

   The inner product of vectors u and v.

2. Norm

3. Orthogonality