Linear Algebra

This content focuses on the general vector spaces within linear algebra, primarily dealing with finite-dimensional linear transformations and inner product spaces. Information on matrices can be found in the Matrix Algebra category. Even if the content is the same, articles in the linear algebra category may be more abstract or challenging.

Vector Spaces

• Definition of Vector Spaces
  • Subspaces
  • Flags
• Linear Combinations, Span
  • The span of the union is the sum of spans $\span(S_{1} \cup S_{2}) = \span(S_{1}) + \span(S_{2})$
• Linear Independence and Dependence
  • Definition of Wronskian and Independence/Dependence Test
  • Affine Independence
• Bases
  • Theorems on Adding/Removing Elements from a Basis
  • Necessary and Sufficient Conditions for Being a Basis in a Finite-Dimensional Vector Space
  • Expansion and Reduction of Bases
  • Ordered Bases and Coordinate Vectors
  • Coordinate Transformation of Vectors $[\mathbf{v}]_{\beta} = Q [\mathbf{v}]_{\beta^{\prime}}$
  • Coordinate Transformation of Linear Transformations $\begin{bmatrix} T \end{bmatrix}{\beta^{\prime}} = Q^{-1} \begin{bmatrix} T \end{bmatrix}{\beta} Q$
• Dimension
• Sum $+$
• Direct Sum $\oplus$
  • Properties of Direct Sum
• Convex Sets
• Matrix Vector Spaces
• Polynomial Vector Spaces
• Definition of Cone and Convex Cone

Quotient Spaces and Cosets

• Cosets and Quotient Spaces $v + W$, $V/W$
• Basis and Dimension of Quotient Spaces
• Mapping to Quotient Spaces $\eta : V \to V/W$
• Linear Transformation on Quotient Spaces $\overline{T} : V/W \to V/W$
• Relationship between Linear Transformations and Mappings to Quotient Spaces in terms of Characteristic Polynomials
• Transformations on Quotient Spaces of Diagonalizable Linear Transformations are also Diagonalizable

Linear Transformations

• Linear Transformations $T : V \to W$
  • Composition
  • Inverse Transformation $T^{-1} : W \to V$
  • Null Space and Range $N(T)$, $R(T)$
  • Rank, Nullity, Dimension Theorem $\dim(N(T)) + \dim(R(T)) = \dim(V)$
  • Norm
  • Matrix Representation $\begin{bmatrix} T \end{bmatrix}_{\beta}$
    • Matrix Representation of Sums and Scalar Multiples
    • Matrix Representation of Linear Transformations with Respect to an Extended Basis from a Subspace
  • Left Multiplication Transformations (Matrix Transformations)
  • Trace $\tr$
• Necessary and Sufficient Conditions for Injectivity and Surjectivity
• The Basis of the Domain Generates the Image of the Linear Transformation
• Linear Transformations between Two Finite-Dimensional Vector Spaces
• Isomorphisms
  • Every $n$-dimensional Real Vector Space is Isomorphic to $\mathbb{R}^{n}$
  • The Space of Linear Transformations and its Matrix Representation Space are Isomorphic
• Spaces of Linear Transformations $L(V,W)$, $\operatorname{Hom}(V,W)$, $\operatorname{End}(V)$
  • Properties of the Space of Invertible Linear Transformations
• Conditions Equivalent to the Range Being Smaller than the Kernel
• Nilpotent $T^{k} = 0$
  • The Only Eigenvalue of a Nilpotent is Zero
  • Properties of the Null Space

Eigenvalues and Diagonalization

• Diagonalizable Linear Transformations
• Eigenvalues, Eigenvectors $Tv = \lambda v$
  • Eigenvectors Corresponding to Different Eigenvalues are Linearly Independent
  • Multiplicity of Eigenvalues
  • Eigenspaces $E_{\lambda}$, Geometric Multiplicity of Eigenvalues
  • The Union of Linearly Independent Sets from Different Eigenspaces is Linearly Independent
  • Relationship between Diagonalizability of a Linear Transformation and the Multiplicity and Eigenspaces of Eigenvalues
• Characteristic Polynomial $f(t) = \det(T - tI)$
  • The Characteristic Polynomial of a Diagonalizable Linear Transformation is Factorizable
  • Cayley-Hamilton Theorem $f(T) = T_{0}$
  • Necessary and Sufficient Conditions for the Factorization of the Characteristic Polynomial
• Invariant Subspaces
  • Relationship between Invariant Subspaces and Eigenvectors
  • Restriction Mappings to Invariant Subspaces of Diagonalizable Linear Transformations are also Diagonalizable
  • Sufficient Conditions for a Linear Transformation to be Diagonalizable
  • Direct Sum of Invariant Subspaces and its Characteristic Polynomial
• Cyclic Subspaces

Integral Transforms

• Integral Transforms
• General Definition of Convolution

Dual Spaces

• Linear Functionals
  • Necessary and Sufficient Conditions for Continuity
  • Necessary and Sufficient Conditions for Representation as a Linear Combination of Independent Functionals
• Linear Forms
• Quadratic Forms
  • Necessary and Sufficient Conditions for a Quadratic Form to be Zero
  • Eigenvalues of Positive-Definite Matrices and the Maximum Value of a Quadratic Form
• Bilinear Forms and Hermitian Forms
• Dual Spaces
• Double Dual Spaces
• Reflexive Spaces
• Transpose of Linear Transformations Defined via Dual Spaces

Tensors

• Tensor Product of Vector Spaces $V \otimes W$
• Product Vectors $v \otimes w$
• Universal Property of Tensor Products
• Tensor Product of Linear Transformations $\phi \otimes \psi$
• Matrix Representation of Tensor Products $\begin{bmatrix} \phi \otimes \psi \end{bmatrix}_{\mathcal{V} \otimes \mathcal{W}}^{{\mathcal{V}}^{\prime} \otimes \mathcal{W}^{\prime}}$
• Tensor Product and Dual Spaces $V^{\ast} \otimes W^{\ast} \cong (V \otimes W)^{\ast}$
• Transpose of Tensor Products $(phi \otimes psi)^{t}$

Inner Product Spaces

• What is an Inner Product in Real Vector Spaces?
• Orthogonal Complements
• Relationship between Orthogonality and Linear Independence
• Coordinates Relative to Orthogonal Bases
• Projection Theorem in Linear Algebra
• Gram-Schmidt Orthogonalization
• What is a Norm in Linear Algebra?
• Equivalence of Norms
• Hölder’s Inequality
• Minkowski’s Inequality

Key References

• Stephen H. Friedberg, Linear Algebra (4th Edition, 2002)
• Howard Anton, Elementary Linear Algebra: Applications Version (12th Edition, 2019)