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
- Linear Combinations, Span
- Linear Independence and Dependence
- 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 $\oplus$
- 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 $T : V \to W$
- 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
- Spaces of Linear Transformations $L(V,W)$, $\operatorname{Hom}(V,W)$, $\operatorname{End}(V)$
- 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
- Cyclic Subspaces
Integral Transforms
Dual Spaces
- Linear Functionals
- Linear Forms
- Quadratic Forms
- 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)
All posts
- Definition of Vector Space
- Subspace of Vector Space
- Linear Combination, Span
- Linear Independence and Linear Dependence
- Dimension of the Vector Space
- Direct Sum in Vector Spaces
- Gram-Schmidt Orthonormalization
- Wronskian Definition and Determination of Linear Independence
- In Linear Algebra, What is a Norm?
- Holder Inequality
- Orthogonal Complement of a Subspace
- Homogeneity of Norms
- Minkowski Inequality
- Necessary and Sufficient Conditions for Linear Functionals to be Continuous
- Necessary and Sufficient Conditions for Linear Functionals to be Represented by Linearly Independent Combinations
- Dual Space
- Reflexive of Vector Spaces
- Integral Transformation
- Convolution's General Definition
- Convex Sets in Vector Spaces
- Linear Forms
- Basis of Vector Space
- Projection Theorem in Linear Algebra
- Necessary and Sufficient Conditions for a Basis in Finite-Dimensional Vector Spaces
- Linear Transformation
- What is Inner Product in Real Vector Spaces?
- Relationship Between Orthogonality and Linear Independence
- Basis Addition/Subtraction Theorem
- Orthogonal Basis and Its Coordinates
- The Basis of the Domain Generates the Image of the Linear Transformation
- Linear Transformation: Kernel and Range
- Rank, Nullity, and Dimension Theorems of Linear Transformations
- Necessary and Sufficient Conditions for Linear Transformations to be Surjective and Injective
- Composition of Linear Transformations
- Norm of Linear Transformations
- Properties of the Space of Invertible Linear Transformations
- The Matrix Representation of a Linear Transformation
- Every n-dimensional Real Vector Space is Isomorphic to R^n
- Linear Transformation Trace
- Linear Transformations Between Finite-Dimensional Vector Spaces
- Order Basis and Coordinate Vectors
- Linear Functional
- Linear Transformation Space
- Inverse of Linear Transformations
- Homomorphism
- Linear Transformation Spaces and Their Matrix Representation Spaces are Isomorphic
- Transpose of Linear Transformations Defined by Dual Spaces
- Dual Pair Spaces
- The Equivalence Condition When the Range of a Linear Transformation is Smaller than the Kernel
- Expansion and Contraction of the Basis
- Matrices of Linear Transformations from a Basis of a Subspace to an Extended Basis
- Matrix Spaces
- Left Multiplication Transformation (Matrix Transformation)
- Coordinate Transformation of Vectors
- Basis Transformation (Coordinate Transformation) of Linear Transformations
- Eigenvalues and Eigenvectors of Finite-Dimensional Linear Transformations
- Characteristics Polynomial of Linear Transformation
- Diagonalizable Linear Transformation
- Eigenvectors Corresponding to Distinct Eigenvalues are Linearly Independent
- Polynomial Vector Spaces
- The Characteristic Polynomial of a Diagonalizable Linear Transformation Is Factorizable
- Definition of Affine Independence
- Multiplicity of Eigenvalues of Linear Transformations
- Eigen Spaces of Linear Transformations and Geometric Multiplicity
- The Union of Linearly Independent Sets from Different Eigenspaces is Linearly Independent
- Invariant Subspaces of Vector Spaces
- The Relationship between Invariant Subspaces and Eigenvectors
- Diagonalizable Linear Transformations Also Diagonalize When Restricted to Invariant Subspaces
- Residual Classes and Quotient Spaces in Linear Algebra
- Basis and Dimension of Quotient Space
- Mapping to the Quotient Space
- Linear Transformations on the Quotient Space
- Characteristics of the Mapping into the Quotient Space via Linear Transformations between Polynomial Relations
- Transformations on the Quotient Space of Diagonalizable Linear Transformations are also Diagonalizable
- Sum of Subspaces in a Vector Space
- Properties of Direct Sum
- The Creation of Unions is Equal to the Sum of Creations
- Diagonalizability of Linear Transformations, Multiplicity of Eigenvalues, and the Relationship with Eigenspaces
- Power Series Linear Transformation
- The Eigenvalues of the Null Matrix are Only Zero
- Null Space of Power Maps
- Cyclic Subspaces of Vector Spaces
- Cayley-Hamilton Theorem
- Direct Sum of Invariant Subspaces and Its Characteristic Polynomial
- What is a Flag in Linear Algebra?
- Tensor Product of Vector Spaces
- Tensor Product of Product Vectors
- Universal Properties of Tensor Products
- Tensor Product of Linear Transformations
- Matrix Representation of Tensor Product
- Matrix Representation of the Sum and Scalar Multiplication of Linear Transformations
- Quadratic Form
- Bilinear Forms and Hermitian Forms
- Necessary and Sufficient Conditions for a Quadratic Form to Be Zero
- Eigenvalues of Positive Definite Matrices and the Maximum Value of Quadratic Forms
- Definition of Cone and Convex Cone