Every definition and theorem in Rob Beezer's A First Course in Linear Algbra
Definitions in green; Theorems in blue; Size indicates the relative number of references; Mouse over to see dependencies;
Columns correspond to chapters, blocks of rows to sections; read top-down and left-to-right
Systems of Linear EquationsVectorsMatricesVector SpacesDeterminantsEigenvaluesLinear TransformationsRepresentations Eigenvalues, Eigenvectors, Representations Row Space of a Matrix is a Subspace Determinant of Matrices of Size Two Composition of Surjective Linear Transformations is Surjective Linear Transformation Nonsingular Matrix Equivalences, Round 2 Column Vector Solution of a System of Linear Equations Matrix Representation of a Multiple of a Linear Transformation Conjugation Respects Matrix Addition Inner Product Computing Rank and Nullity Left Null Space Linear Transformation Scalar Multiplication Matrix Rank Of a Matrix Span of a Set Column Space, Row Space, Transpose Similar Matrices have Equal Eigenvalues Matrix of a Linear Transformation, Column Vectors Nonsingular Matrices have Linearly Independent Columns Norm of a Vector Basis for Null Spaces Complex Conjugate of a Matrix Square Matrix Nonsingular Matrix Equivalences, Round 6 Goldilocks Trivial Solution to Homogeneous Systems of Equations Row Space of a Matrix Adjoint and Inner Product Equation Operations Determinant with Equal Rows or Columns Two-by-Two Matrix Inverse Reduced Row-Echelon Form is Unique Row-Equivalent Matrix in Echelon Form Zero Vector is Unique Singular Matrices have Zero Eigenvalues Orthogonal Set of Vectors Equal Matrices and Matrix-Vector Products Characteristic Polynomial Linear Transformations Take Zero to Zero Zero Scalar in Scalar Multiplication Singular Matrices have Zero Determinants Vector Representation is Surjective Eigenspace for a Matrix is a Subspace Normal Matrix Linear Combination Similarity and Change of Basis Orthogonal Vectors Nonsingular Matrices and Unique Solutions Scalar Multiplication Equals the Zero Vector Linear Combination of Column Vectors Range and Column Space Isomorphism Matrix Multiplication Eigenvalues of Real Matrices come in Conjugate Pairs Nonsingular Matrix Equivalences, Round 4 Composition of Injective Linear Transformations is Injective Isomorphism of Finite Dimensional Vector Spaces Coordinatization and Linear Independence Matrix Multiplication and the Zero Matrix Nonsingular Matrices are Products of Elementary Matrices Orthogonal Sets are Linearly Independent Rank Plus Nullity is Domain Dimension Columns of Unitary Matrices are Orthonormal Sets Nonsingular Matrix Equivalences, Round 7 Diagonal Matrix Distinct Eigenvalues implies Diagonalizable Socks and Shoes Determinant of the Transpose Vector Representation is Injective Matrix Multiplication and Scalar Matrix Multiplication Hermitian Matrices and Inner Products Determinant for Row or Column Multiples Kernel of an Injective Linear Transformation Zero Column Vector Matrices Build Linear Transformations Characterization of Finite Dimensional Vector Spaces Nullity Of a Linear Transformation Coordinates and Orthonormal Bases Rank Plus Nullity is Columns Reduced Row-Echelon Form Null Space of a Matrix Surjective Linear Transformation Nonsingular Product has Nonsingular Terms Equal Dimensions Yields Equal Subspaces Product of Triangular Matrices is Triangular Fundamental Theorem of Matrix Representation Vector Space of $m\times n$ Matrices Vector Space Properties of Matrices Nonsingularity is Invertibility Matrix Multiplication and Inner Products Linear Transformation Addition Invertible Linear Transformations are Injective and Surjective Coordinatization and Spanning Sets Eigenvalue and Eigenvector of a Linear Transformation Determinant of the Identity Matrix Transpose of a Transpose Conjugation Respects Vector Addition Properties of Extended Echelon Form Nonsingular Matrices Row Reduce to the Identity matrix Four Subsets Homogeneous Systems are Consistent Matrix Representation of a Composition of Linear Transformations Determinant for Row or Column Multiples and Addition Eigenvalues of the Transpose of a Matrix Maximum Number of Eigenvalues of a Matrix Nonsingular Matrix Equivalences, Round 3 Matrix Representation and Change of Basis Matrix Conjugation and Transposes Zero Matrix Matrix Scalar Multiplication Multiplicities of an Eigenvalue SubMatrix Unitary Matrices Preserve Inner Products Range and Pre-Image Matrix Multiplication and Adjoints Change-of-Basis Augmented Matrix Conjugation Respects Matrix Scalar Multiplication Column Vector Addition Diagonalization Characterization Basis of the Column Space Identity Linear Transformation Injective Linear Transformations and Linear Independence Determinant Expansion about Columns Dependency in Linearly Dependent Sets Hermitian Matrices have Orthogonal Eigenvectors Testing Subsets for Subspaces Rank Of a Surjective Linear Transformation Entries of Matrix Products Similar Matrices Determinant for Row or Column Swap Elementary Matrices Do Row Operations Composition of Invertible Linear Transformations Vector Representation is a Linear Transformation Standard Unit Vectors are a Basis Range of a Surjective Linear Transformation Eigenvalues and Eigenvectors of a Matrix Kernel and Null Space Isomorphism Homogeneous, More Variables than Equations, Infinite solutions Nonsingular Matrix Equivalences, Round 8 Every Matrix Has an Eigenvalue Eigenvalues Of Matrix Powers Row Operations Nonsingular Matrix Equivalences, Round 9 Consistent, More Variables than Equations, Infinite solutions Standard Unit Vectors Vector of Constants System of Linear Equations Equation Operations Preserve Solution Sets Eigenvectors with Distinct Eigenvalues are Linearly Independent Consistent Systems, $r$ and $n$ Orthonormal Basis for Upper Triangular Representation Vector Form of Solutions to Linear Systems Transpose and Matrix Scalar Multiplication Matrix Multiplication Distributes Across Addition Orthonormal Bases and Normal Matrices Linear Independence Inverse of a Composition of Linear Transformations Nonsingular Matrix Equivalences, Round 1 Coefficient Matrix Null Space of a Matrix is a Subspace Trivial Subspaces Inner Product is Anti-Commutative Determinant Respects Matrix Multiplication Spanning Sets and Linear Dependence Orthonormal Diagonalization Nullity Of an Injective Linear Transformation Unitary Matrices are Invertible Dimensions of Four Subspaces Span of a Set is a Subspace Kernel of a Linear Transformation Row-Equivalent Matrices Basis of a Span Vector Space Computing the Inverse of a Nonsingular Matrix Matrix Equality Adjoint and Matrix Scalar Multiplication Composition of Linear Transformations is a Linear Transformation Zero Vector in Scalar Multiplication Dimension of $\complex{m}$ Vector Space of Linear Transformations Kernel and Pre-Image Conjugate of the Conjugate of a Matrix Linearly Independent Vectors, $r$ and $n$ Linear Transformation Defined on a Basis Relation of Linear Dependence for Column Vectors Eigenvalues of a Matrix are Roots of Characteristic Polynomials Solution with Nonsingular Coefficient Matrix Column Vector Equality Range of a Linear Transformation Nonsingular Matrix Equivalences, Round 5 Vector Representation Columns of Nonsingular Matrix are a Basis Spanning Sets for Null Spaces Symmetric Matrices are Square Homogeneous System Degree of the Characteristic Polynomial Hermitian Matrices have Real Eigenvalues Linear Transformation Composition Algebraic Multiplicity of an Eigenvalue Matrix Inverse of a Scalar Multiple Consistent System Row-Equivalent Matrices represent Equivalent Systems Transpose and Matrix Addition Linearly Independent Vectors and Homogeneous Systems Positive Inner Products Matrix Representation of a Sum of Linear Transformations Rank and Nullity of a Nonsingular Matrix More Vectors than Size implies Linear Dependence Possible Solution Sets for Linear Systems Column Space of a Matrix is a Subspace Span of a Set of Column Vectors Nonsingular Matrix Upper Triangular Matrix Transpose of a Matrix Dimension of $P_n$ Invertible Matrices, Invertible Linear Transformation Adjoint of an Adjoint Surjective Linear Transformations and Dimension Matrix Inverse Pre-Image Systems of Linear Equations as Matrix Multiplication Solutions to Linear Systems are Linear Combinations Determinant with Zero Row or Column Inverse of a Triangular Matrix is Triangular Inner Products and Norms Eigenvalues of the Inverse of a Matrix Dimension Sum of Linear Transformations is a Linear Transformation Particular Solution Plus Homogeneous Solutions Nonsingular Matrices have Trivial Null Spaces Injective Linear Transformation Elementary Matrices are Nonsingular Matrix Multiplication and Complex Conjugation Invertible Matrix Representations Subspace Independent and Dependent Variables Inverse of an Invertible Linear Transformation Elementary Matrices Kernel of a Linear Transformation is a Subspace Additive Inverses are Unique Equivalent Systems Proper Subspaces have Smaller Dimension Lower Triangular Matrix Determinants of Elementary Matrices Matrix Multiplication and Identity Matrix Free Variables for Consistent Systems Recognizing Consistency of a Linear System Linear Independence of Column Vectors Unitary Matrices Hermitian Matrix Extending Linearly Independent Sets Complex Conjugate of a Column Vector Inverse of Change-of-Basis Matrix Matrix Inverse of a Transpose Gram-Schmidt Procedure Geometric Multiplicity of an Eigenvalue Rank of a Matrix is the Rank of the Transpose Matrix Multiplication is Associative Bases have Identical Sizes Column Spaces and Consistent Systems Vector Space Properties of Column Vectors Column Space of a Nonsingular Matrix Basis for the Row Space Eigenvalues of a Scalar Multiple of a Matrix Matrix Representation Matrix-Vector Product Eigenvalues of the Polynomial of a Matrix Basis Upper Triangular Matrix Representation Solution Vector Injective Linear Transformations and Bases Number of Eigenvalues of a Matrix Extended Echelon Form Rank Of a Linear Transformation Solution Set of a System of Linear Equations Symmetric Matrix Surjective Linear Transformations and Bases Spanning Set for Range of a Linear Transformation Isomorphic Vector Spaces Injective Linear Transformations and Dimension Matrix Inverse is Unique Determinant Expansion about Rows Unitary Matrices Convert Orthonormal Bases Determinant of a Matrix Linear Transformations and Linear Combinations Nullity Of a Matrix Eigenspace of a Matrix Conjugation Respects Vector Scalar Multiplication Multiple of a Linear Transformation is a Linear Transformation Adjoint Identity Matrix Column Vector Scalar Multiplication Matrix Multiplication and Transposes Relation of Linear Dependence Diagonalizable Matrix Row-Equivalent Matrices have equal Row Spaces Vector Representation is an Invertible Linear Transformation Invertible Linear Transformations Column Space of a Matrix Matrix Representation of a Linear System Eigenspace of a Matrix is a Null Space Change-of-Basis Matrix Additive Inverses from Scalar Multiplication Range of a Linear Transformation is a Subspace Determinants, Elementary Matrices, Matrix Multiplication Similarity is an Equivalence Relation OrthoNormal Set Left Null Space of a Matrix is a Subspace Vector Representation Relative to a Basis Spanning Set of a Vector Space Matrix Addition Inverse of a Linear Transformation is a Linear Transformation Dimension of $M_{mn}$ Vector Space of Column Vectors Diagonalizable Matrices have Full Eigenspaces Matrix Inverse of a Matrix Inverse One-Sided Inverse is Sufficient Isomorphic Vector Spaces have Equal Dimension