Vector Space Notation

Basic notation for vector spaces and their elements:

1. Common Vector Spaces

   \mathbb{R}^n, \mathbb{C}^n, \mathbb{F}^n

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

2. Vector Space Membership

   \mathbf{v} \in V, \quad V \subset \mathbb{R}^n

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

3. Function Spaces

   C[a,b], \quad L^2(\Omega), \quad H^1(\Omega)

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

Basis and Dimension

Representing basis vectors and dimensions:

1. Standard Basis

   \{\mathbf{e}_1, \mathbf{e}_2, \ldots, \mathbf{e}_n\}

   The standard basis for ℝⁿ.

2. General Basis

   \mathcal{B} = \{\mathbf{v}_1, \mathbf{v}_2, \ldots, \mathbf{v}_n\}

   A general basis for an n-dimensional vector space.

3. Dimension

   \dim(V) = n

   The dimension of a vector space V.

Linear Combinations and Span

1. Linear Combination

   c_1\mathbf{v}_1 + c_2\mathbf{v}_2 + \cdots + c_n\mathbf{v}_n

   A linear combination of vectors.

2. Span

   \ ext{span}\{\mathbf{v}_1, \mathbf{v}_2, \ldots, \mathbf{v}_n\}

   The span of a set of vectors.

3. Linear Independence

   c_1\mathbf{v}_1 + c_2\mathbf{v}_2 + \cdots + c_n\mathbf{v}_n = \mathbf{0} \implies c_1 = c_2 = \cdots = c_n = 0

   The definition of linear independence.

Subspaces and Direct Sums

1. Subspace

   U \leq V

   U is a subspace of V.

2. Direct Sum

   V = U \oplus W

   V is the direct sum of subspaces U and W.

3. Null Space and Range

   \ ext{Null}(A) = \{\mathbf{x} \in \mathbb{R}^n : A\mathbf{x} = \mathbf{0}\}

   \ ext{Range}(A) = \{A\mathbf{x} : \mathbf{x} \in \mathbb{R}^n\}

Linear Transformations

1. Linear Transformation Definition

   T: V \ o W

   A linear transformation from vector space V to W.

2. Matrix Representation

   [T]_{\mathcal{B}}^{\mathcal{C}} = \begin{pmatrix} a_{11} & a_{12} & \cdots & a_{1n} \\ a_{21} & a_{22} & \cdots & a_{2n} \\ \vdots & \vdots & \ddots & \vdots \\ a_{m1} & a_{m2} & \cdots & a_{mn} \end{pmatrix}

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

3. Composition of Linear Transformations

   (S \circ T)(\mathbf{v}) = S(T(\mathbf{v}))

Eigenvalues and Eigenvectors

1. Eigenvalue Equation

   A\mathbf{v} = \lambda\mathbf{v}

   The defining equation for eigenvalue λ and eigenvector v.

2. Characteristic Polynomial

   p_A(\lambda) = \det(A - \lambda I)

3. Eigenspace

   E_\lambda = \{\mathbf{v} \in V : A\mathbf{v} = \lambda\mathbf{v}\}

   The eigenspace corresponding to eigenvalue λ.

Inner Product Spaces

1. Inner Product

   \langle \mathbf{u}, \mathbf{v} \rangle

   The inner product of vectors u and v.

2. Norm

   \|\mathbf{v}\| = \sqrt{\langle \mathbf{v}, \mathbf{v} \rangle}

3. Orthogonality

   \mathbf{u} \perp \mathbf{v} \iff \langle \mathbf{u}, \mathbf{v} \rangle = 0