Theorems

Section WILA

Section SSLE

Theorem EOPSS Equation Operations Preserve Solution Sets

Section RREF

Theorem REMES Row-Equivalent Matrices represent Equivalent Systems

Theorem REMEF Row-Equivalent Matrix in Echelon Form

Section TSS

Theorem RCLS Recognizing Consistency of a Linear System

Theorem ISRN Inconsistent Systems, $r$
and $n$

Theorem CSRN Consistent Systems, $r$
and $n$

Theorem FVCS Free Variables for Consistent Systems

Theorem PSSLS Possible Solution Sets for Linear Systems

Theorem CMVEI Consistent, More Variables than Equations, Infinite
solutions

Section HSE

Theorem HSC Homogeneous Systems are Consistent

Theorem HMVEI Homogeneous, More Variables than Equations, Infinite
solutions

Section NM

Theorem NMRRI Nonsingular Matrices Row Reduce to the Identity matrix

Theorem NMTNS Nonsingular Matrices have Trivial Null Spaces

Theorem NMUS Nonsingular Matrices and Unique Solutions

Theorem NME1 Nonsingular Matrix Equivalences, Round 1

Section VO

Theorem VSPCV Vector Space Properties of Column Vectors

Section LC

Theorem SLSLC Solutions to Linear Systems are Linear Combinations

Theorem VFSLS Vector Form of Solutions to Linear Systems

Theorem PSPHS Particular Solution Plus Homogeneous Solutions

Theorem RREFU Reduced Row-Echelon Form is Unique

Section SS

Theorem SSNS Spanning Sets for Null Spaces

Section LI

Theorem LIVHS Linearly Independent Vectors and Homogeneous Systems

Theorem LIVRN Linearly Independent Vectors,
$r$ and
$n$

Theorem MVSLD More Vectors than Size implies Linear Dependence

Theorem NMLIC Nonsingular Matrices have Linearly Independent Columns

Theorem NME2 Nonsingular Matrix Equivalences, Round 2

Theorem BNS Basis for Null Spaces

Section LDS

Theorem DLDS Dependency in Linearly Dependent Sets

Theorem BS Basis of a Span

Section O

Theorem CRVA Conjugation Respects Vector Addition

Theorem CRSM Conjugation Respects Vector Scalar Multiplication

Theorem IPVA Inner Product and Vector Addition

Theorem IPSM Inner Product and Scalar Multiplication

Theorem IPAC Inner Product is Anti-Commutative

Theorem IPN Inner Products and Norms

Theorem PIP Positive Inner Products

Theorem OSLI Orthogonal Sets are Linearly Independent

Theorem GSP Gram-Schmidt Procedure

Section MO

Theorem VSPM Vector Space Properties of Matrices

Theorem SMS Symmetric Matrices are Square

Theorem TMA Transpose and Matrix Addition

Theorem TMSM Transpose and Matrix Scalar Multiplication

Theorem TT Transpose of a Transpose

Theorem CRMA Conjugation Respects Matrix Addition

Theorem CRMSM Conjugation Respects Matrix Scalar Multiplication

Theorem CCM Conjugate of the Conjugate of a Matrix

Theorem MCT Matrix Conjugation and Transposes

Theorem AMA Adjoint and Matrix Addition

Theorem AMSM Adjoint and Matrix Scalar Multiplication

Theorem AA Adjoint of an Adjoint

Section MM

Theorem SLEMM Systems of Linear Equations as Matrix Multiplication

Theorem EMMVP Equal Matrices and Matrix-Vector Products

Theorem EMP Entries of Matrix Products

Theorem MMZM Matrix Multiplication and the Zero Matrix

Theorem MMIM Matrix Multiplication and Identity Matrix

Theorem MMDAA Matrix Multiplication Distributes Across Addition

Theorem MMSMM Matrix Multiplication and Scalar Matrix Multiplication

Theorem MMA Matrix Multiplication is Associative

Theorem MMIP Matrix Multiplication and Inner Products

Theorem MMCC Matrix Multiplication and Complex Conjugation

Theorem MMT Matrix Multiplication and Transposes

Theorem MMAD Matrix Multiplication and Adjoints

Theorem AIP Adjoint and Inner Product

Theorem HMIP Hermitian Matrices and Inner Products

Section MISLE

Theorem TTMI Two-by-Two Matrix Inverse

Theorem CINM Computing the Inverse of a Nonsingular Matrix

Theorem MIU Matrix Inverse is Unique

Theorem SS Socks and Shoes

Theorem MIMI Matrix Inverse of a Matrix Inverse

Theorem MIT Matrix Inverse of a Transpose

Theorem MISM Matrix Inverse of a Scalar Multiple

Section MINM

Theorem NPNT Nonsingular Product has Nonsingular Terms

Theorem OSIS One-Sided Inverse is Sufficient

Theorem NI Nonsingularity is Invertibility

Theorem NME3 Nonsingular Matrix Equivalences, Round 3

Theorem SNCM Solution with Nonsingular Coefficient Matrix

Theorem UMI Unitary Matrices are Invertible

Theorem CUMOS Columns of Unitary Matrices are Orthonormal Sets

Theorem UMPIP Unitary Matrices Preserve Inner Products

Section CRS

Theorem CSCS Column Spaces and Consistent Systems

Theorem BCS Basis of the Column Space

Theorem CSNM Column Space of a Nonsingular Matrix

Theorem NME4 Nonsingular Matrix Equivalences, Round 4

Theorem REMRS Row-Equivalent Matrices have equal Row Spaces

Theorem BRS Basis for the Row Space

Theorem CSRST Column Space, Row Space, Transpose

Section FS

Theorem PEEF Properties of Extended Echelon Form

Theorem FS Four Subsets

Section VS

Theorem ZVU Zero Vector is Unique

Theorem AIU Additive Inverses are Unique

Theorem ZSSM Zero Scalar in Scalar Multiplication

Theorem ZVSM Zero Vector in Scalar Multiplication

Theorem AISM Additive Inverses from Scalar Multiplication

Theorem SMEZV Scalar Multiplication Equals the Zero Vector

Section S

Theorem TSS Testing Subsets for Subspaces

Theorem NSMS Null Space of a Matrix is a Subspace

Theorem SSS Span of a Set is a Subspace

Theorem CSMS Column Space of a Matrix is a Subspace

Theorem RSMS Row Space of a Matrix is a Subspace

Theorem LNSMS Left Null Space of a Matrix is a Subspace

Section LISS

Theorem VRRB Vector Representation Relative to a Basis

Section B

Theorem SUVB Standard Unit Vectors are a Basis

Theorem CNMB Columns of Nonsingular Matrix are a Basis

Theorem NME5 Nonsingular Matrix Equivalences, Round 5

Theorem COB Coordinates and Orthonormal Bases

Theorem UMCOB Unitary Matrices Convert Orthonormal Bases

Section D

Theorem SSLD Spanning Sets and Linear Dependence

Theorem BIS Bases have Identical Sizes

Theorem DCM Dimension of ${\u2102}^{m}$

Theorem DP Dimension of ${P}_{n}$

Theorem DM Dimension of ${M}_{mn}$

Theorem CRN Computing Rank and Nullity

Theorem RPNC Rank Plus Nullity is Columns

Theorem RNNM Rank and Nullity of a Nonsingular Matrix

Theorem NME6 Nonsingular Matrix Equivalences, Round 6

Section PD

Theorem ELIS Extending Linearly Independent Sets

Theorem G Goldilocks

Theorem PSSD Proper Subspaces have Smaller Dimension

Theorem EDYES Equal Dimensions Yields Equal Subspaces

Theorem RMRT Rank of a Matrix is the Rank of the Transpose

Theorem DFS Dimensions of Four Subspaces

Theorem DSFB Direct Sum From a Basis

Theorem DSFOS Direct Sum From One Subspace

Theorem DSZV Direct Sums and Zero Vectors

Theorem DSZI Direct Sums and Zero Intersection

Theorem DSLI Direct Sums and Linear Independence

Theorem DSD Direct Sums and Dimension

Theorem RDS Repeated Direct Sums

Section DM

Theorem EMDRO Elementary Matrices Do Row Operations

Theorem EMN Elementary Matrices are Nonsingular

Theorem NMPEM Nonsingular Matrices are Products of Elementary
Matrices

Theorem DMST Determinant of Matrices of Size Two

Theorem DER Determinant Expansion about Rows

Theorem DT Determinant of the Transpose

Theorem DEC Determinant Expansion about Columns

Section PDM

Theorem DZRC Determinant with Zero Row or Column

Theorem DRCS Determinant for Row or Column Swap

Theorem DRCM Determinant for Row or Column Multiples

Theorem DERC Determinant with Equal Rows or Columns

Theorem DRCMA Determinant for Row or Column Multiples and Addition

Theorem DIM Determinant of the Identity Matrix

Theorem DEM Determinants of Elementary Matrices

Theorem DEMMM Determinants, Elementary Matrices, Matrix Multiplication

Theorem SMZD Singular Matrices have Zero Determinants

Theorem NME7 Nonsingular Matrix Equivalences, Round 7

Theorem DRMM Determinant Respects Matrix Multiplication

Section EE

Theorem EMHE Every Matrix Has an Eigenvalue

Theorem EMRCP Eigenvalues of a Matrix are Roots of Characteristic
Polynomials

Theorem EMS Eigenspace for a Matrix is a Subspace

Theorem EMNS Eigenspace of a Matrix is a Null Space

Section PEE

Theorem EDELI Eigenvectors with Distinct Eigenvalues are Linearly
Independent

Theorem SMZE Singular Matrices have Zero Eigenvalues

Theorem NME8 Nonsingular Matrix Equivalences, Round 8

Theorem ESMM Eigenvalues of a Scalar Multiple of a Matrix

Theorem EOMP Eigenvalues Of Matrix Powers

Theorem EPM Eigenvalues of the Polynomial of a Matrix

Theorem EIM Eigenvalues of the Inverse of a Matrix

Theorem ETM Eigenvalues of the Transpose of a Matrix

Theorem ERMCP Eigenvalues of Real Matrices come in Conjugate Pairs

Theorem DCP Degree of the Characteristic Polynomial

Theorem NEM Number of Eigenvalues of a Matrix

Theorem ME Multiplicities of an Eigenvalue

Theorem MNEM Maximum Number of Eigenvalues of a Matrix

Theorem HMRE Hermitian Matrices have Real Eigenvalues

Theorem HMOE Hermitian Matrices have Orthogonal Eigenvectors

Section SD

Theorem SER Similarity is an Equivalence Relation

Theorem SMEE Similar Matrices have Equal Eigenvalues

Theorem DC Diagonalization Characterization

Theorem DMFE Diagonalizable Matrices have Full Eigenspaces

Theorem DED Distinct Eigenvalues implies Diagonalizable

Section LT

Theorem LTTZZ Linear Transformations Take Zero to Zero

Theorem MBLT Matrices Build Linear Transformations

Theorem MLTCV Matrix of a Linear Transformation, Column Vectors

Theorem LTLC Linear Transformations and Linear Combinations

Theorem LTDB Linear Transformation Defined on a Basis

Theorem SLTLT Sum of Linear Transformations is a Linear Transformation

Theorem MLTLT Multiple of a Linear Transformation is a Linear Transformation

Theorem VSLT Vector Space of Linear Transformations

Theorem CLTLT Composition of Linear Transformations is a Linear
Transformation

Section ILT

Theorem KLTS Kernel of a Linear Transformation is a Subspace

Theorem KPI Kernel and Pre-Image

Theorem KILT Kernel of an Injective Linear Transformation

Theorem ILTLI Injective Linear Transformations and Linear Independence

Theorem ILTB Injective Linear Transformations and Bases

Theorem ILTD Injective Linear Transformations and Dimension

Theorem CILTI Composition of Injective Linear Transformations is Injective

Section SLT

Theorem RLTS Range of a Linear Transformation is a Subspace

Theorem RSLT Range of a Surjective Linear Transformation

Theorem SSRLT Spanning Set for Range of a Linear Transformation

Theorem RPI Range and Pre-Image

Theorem SLTB Surjective Linear Transformations and Bases

Theorem SLTD Surjective Linear Transformations and Dimension

Theorem CSLTS Composition of Surjective Linear Transformations is
Surjective

Section IVLT

Theorem ILTLT Inverse of a Linear Transformation is a Linear Transformation

Theorem IILT Inverse of an Invertible Linear Transformation

Theorem ILTIS Invertible Linear Transformations are Injective and Surjective

Theorem CIVLT Composition of Invertible Linear Transformations

Theorem ICLT Inverse of a Composition of Linear Transformations

Theorem IVSED Isomorphic Vector Spaces have Equal Dimension

Theorem ROSLT Rank Of a Surjective Linear Transformation

Theorem NOILT Nullity Of an Injective Linear Transformation

Theorem RPNDD Rank Plus Nullity is Domain Dimension

Section VR

Theorem VRLT Vector Representation is a Linear Transformation

Theorem VRI Vector Representation is Injective

Theorem VRS Vector Representation is Surjective

Theorem VRILT Vector Representation is an Invertible Linear Transformation

Theorem CFDVS Characterization of Finite Dimensional Vector Spaces

Theorem IFDVS Isomorphism of Finite Dimensional Vector Spaces

Theorem CLI Coordinatization and Linear Independence

Theorem CSS Coordinatization and Spanning Sets

Section MR

Theorem FTMR Fundamental Theorem of Matrix Representation

Theorem MRSLT Matrix Representation of a Sum of Linear Transformations

Theorem MRMLT Matrix Representation of a Multiple of a Linear
Transformation

Theorem MRCLT Matrix Representation of a Composition of Linear
Transformations

Theorem KNSI Kernel and Null Space Isomorphism

Theorem RCSI Range and Column Space Isomorphism

Theorem IMR Invertible Matrix Representations

Theorem IMILT Invertible Matrices, Invertible Linear Transformation

Theorem NME9 Nonsingular Matrix Equivalences, Round 9

Section CB

Theorem CB Change-of-Basis

Theorem ICBM Inverse of Change-of-Basis Matrix

Theorem CBOB Change of Basis for Orthonormal Bases

Theorem MRCB Matrix Representation and Change of Basis

Theorem SCB Similarity and Change of Basis

Theorem EER Eigenvalues, Eigenvectors, Representations

Section OD

Theorem PTMT Product of Triangular Matrices is Triangular

Theorem ITMT Inverse of a Triangular Matrix is Triangular

Theorem UTMR Upper Triangular Matrix Representation

Theorem OBUTR Orthonormal Basis for Upper Triangular Representation

Theorem OD Orthonormal Diagonalization

Theorem OBNM Orthonormal Bases and Normal Matrices

Section NLT

Theorem NJB Nilpotent Jordan Blocks

Theorem ENLT Eigenvalues of Nilpotent Linear Transformations

Theorem DNLT Diagonalizable Nilpotent Linear Transformations

Theorem KPLT Kernels of Powers of Linear Transformations

Theorem KPNLT Kernels of Powers of Nilpotent Linear Transformations

Theorem CFNLT Canonical Form for Nilpotent Linear Transformations

Section IS

Theorem EIS Eigenspaces are Invariant Subspaces

Theorem KPIS Kernels of Powers are Invariant Subspaces

Theorem GESIS Generalized Eigenspace is an Invariant Subspace

Theorem GEK Generalized Eigenspace as a Kernel

Theorem RGEN Restriction to Generalized Eigenspace is Nilpotent

Theorem MRRGE Matrix Representation of a Restriction to a Generalized
Eigenspace

Section JCF

Theorem GESD Generalized Eigenspace Decomposition

Theorem DGES Dimension of Generalized Eigenspaces

Theorem JCFLT Jordan Canonical Form for a Linear Transformation

Theorem CHT Cayley-Hamilton Theorem

Section CNO

Theorem PCNA Properties of Complex Number Arithmetic

Theorem CCRA Complex Conjugation Respects Addition

Theorem CCRM Complex Conjugation Respects Multiplication

Theorem CCT Complex Conjugation Twice

Section SET

Section PT

Section F

Theorem FIMP Field of Integers Modulo a Prime

Section T

Theorem TL Trace is Linear

Theorem TSRM Trace is Symmetric with Respect to Multiplication

Theorem TIST Trace is Invariant Under Similarity Transformations

Theorem TSE Trace is the Sum of the Eigenvalues

Section VM

Theorem DVM Determinant of a Vandermonde Matrix

Theorem NVM Nonsingular Vandermonde Matrix

Section PSM

Theorem CPSM Creating Positive Semi-Definite Matrices

Theorem EPSM Eigenvalues of Positive Semi-definite Matrices

Section ROD

Theorem ROD Rank One Decomposition

Section TD

Theorem TD Triangular Decomposition

Theorem TDEE Triangular Decomposition, Entry by Entry

Section SVD

Theorem EEMAP Eigenvalues and Eigenvectors of Matrix-Adjoint Product

Theorem SVD Singular Value Decomposition

Section SR

Theorem PSMSR Positive Semi-Definite Matrices and Square Roots

Theorem EESR Eigenvalues and Eigenspaces of a Square Root

Theorem USR Unique Square Root

Section POD

Theorem PDM Polar Decomposition of a Matrix

Section CF

Theorem IP Interpolating Polynomial

Theorem LSMR Least Squares Minimizes Residuals

Section SAS