Theorems

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 GSPCV Gram-Schmidt Procedure, Column Vectors
 
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
Theorem VAC Vector Addition Cancellation
Theorem CSSM Canceling Scalars in Scalar Multiplication
Theorem CVSM Canceling Vectors in Scalar Multiplication
 
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
 
Section D
Theorem SSLD Spanning Sets and Linear Dependence
Theorem BIS Bases have Identical Sizes
Theorem DCM Dimension of ${ℂ}^m$
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 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 UTMR Upper Triangular Matrix Representation
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 ROD
Theorem ROD Rank One Decomposition
 
Section TD
Theorem TD Triangular Decomposition
Theorem TDEE Triangular Decomposition, Entry by Entry