Appendix B. List of Theorems
1 Linear Equations in Linear Algebra
Theorem 1.20 Existence and Uniqueness of Solutions
Theorem 1.36 Equivalent Formulations of a Linear System
Theorem 1.37 The Existence of Solutions for the Matrix Equation
Theorem 1.38 Consistency for all \(\vec{b}\) and pivots
Theorem 1.46 The Zero Vector and Linear Dependence
Theorem 1.47 Too Many Vectors and Linear Dependence
Theorem 1.58 The Standard Matrix of a Linear Transformation
Theorem 1.61 Onto: Equivalent Statements
Theorem 1.63 One-to-One: Equivalent Statements
2 Matrix Algebra
Theorem 2.5 Row-Column Rule for Matrix Multiplication
Theorem 2.7 Properties and Warnings
Theorem 2.11 Properties of Matrix Transpose
Theorem 2.17 Inverse Formula for a \(2\times 2\) Matrix
Theorem 2.19 Inverse and Unique Solutions
Theorem 2.25 Invertibility and Row Equivalence
Theorem 2.26 The Invertible Matrix Theorem (IMT)
Theorem 2.27 Left and Right Inverses
Theorem 2.30 \(LU\) Factorization
Theorem 2.40 Subspace and Span
Theorem 2.42 Null Space and Column Space of Row Equivalent Matrices
Theorem 2.54 Dimension Is Well-Defined
Theorem 2.58 Rank Theorem
Theorem 2.59 Basis Theorem
Theorem 2.60 The Invertible Matrix Theorem (Continued)
3 Determinants
Theorem 3.6 Cofactor Expansion along Any Row or Column
Theorem 3.8 Determinant of a Triangular Matrix
Theorem 3.10 Row Operations and the Determinant
Theorem 3.13 Determinant after Row Replacements
Theorem 3.14 Propertise of the Determinant
Theorem 3.17 Area from the Determinant
Theorem 3.18 Volume from the Determinant
Theorem 3.21 Volume Scaling by a Linear Transformation
Theorem 3.23 Cramer’s Rule
4 Vector Spaces
Theorem 4.6 Convergence Theorem
5 Eigenvectors and Eigenvalues
Theorem 5.6 Eigenvalues of Triangular Matrices
Theorem 5.7 Invertibility and the Eigenvalue 0
Theorem 5.8 Stochastic Matrices
Theorem 5.9 Distinct Eigenvalues Give Independent Eigenvectors
Theorem 5.18 Determinant, Trace, and Eigenvalues
Theorem 5.28 Characteristic Polynomial of Similar Matrices
Theorem 5.33 Diagonalization Criterion
Theorem 5.34 Algebraic vs. Geometric Multiplicity
Theorem 5.35 Diagonalization with Distinct Eigenvalues
Theorem 5.48 Conjugate Properties
Theorem 5.49 Fundamental Theorem of Algebra
Theorem 5.50 Conjugate Pairs for Real Coefficients
6 Orthogonality and Least Squares
Theorem 6.3 Basic Identities of the Dot Product
Theorem 6.17 The Four Subspaces
Theorem 6.18 Dot Product and Angle
Theorem 6.22 Linear Independence of Orthogonal Sets
Theorem 6.23 Expansion in an Orthogonal Basis
Theorem 6.26 Properties of Orthogonal Projections
Theorem 6.30 Expansion in an Orthonormal Basis
Theorem 6.32 Orthonormal Columns
Theorem 6.34 Mapping Properties of Matrices with Orthonormal Columns
Theorem 6.36 Orthogonal Decomposition
Theorem 6.39 Best Approximation Theorem
Theorem 6.44 The Gram–Schmidt Process
Theorem 6.47 QR Factorization
Theorem 6.54 Normal Equations for Least Squares
Theorem 6.56 Unique Least Squares Solutions
Theorem 6.58 Least Squares via QR Factorization
7 Google PageRank, Symmetric Matrices, and Quadratic Forms
Theorem 7.3 Convergence of Regular Stochastic Matrices
Theorem 7.15 Orthogonality of Eigenspaces
Theorem 7.18 Spectral Theorem
Theorem 7.19 Spectral Decomposition
Theorem 7.26 Principal Axes Theorem
Theorem 7.29 Eigenvalue Classification
Theorem 7.33 Extrema via Eigenvalues
Theorem 7.35 Extrema with an Orthogonality Condition
Theorem 7.37 Singular Value Decomposition
Theorem 7.42 SVD and the Four Subspaces
Theorem 7.44 Spectral Decomposition
