Section WILA: What is Linear Algebra?
Section SSLE: Solving Systems of Linear Equations
EOPSS Equation Operations Preserve Solution Sets
Section RREF: Reduced Row-Echelon Form
REMES Row-Equivalent Matrices represent Equivalent Systems
REMEF Row-Equivalent Matrix in Echelon Form
RREFU Reduced Row-Echelon Form is Unique
Section TSS: Types of Solution Sets
RCLS Recognizing Consistency of a Linear System
CSRN Consistent Systems, $r$ and $n$
FVCS Free Variables for Consistent Systems
PSSLS Possible Solution Sets for Linear Systems
CMVEI Consistent, More Variables than Equations, Infinite solutions
Section HSE: Homogeneous Systems of Equations
HSC Homogeneous Systems are Consistent
HMVEI Homogeneous, More Variables than Equations, Infinite solutions
Section NM: Nonsingular Matrices
NMRRI Nonsingular Matrices Row Reduce to the Identity matrix
NMTNS Nonsingular Matrices have Trivial Null Spaces
NMUS Nonsingular Matrices and Unique Solutions
NME1 Nonsingular Matrix Equivalences, Round 1
Section VO: Vector Operations
VSPCV Vector Space Properties of Column Vectors
Section LC: Linear Combinations
SLSLC Solutions to Linear Systems are Linear Combinations
VFSLS Vector Form of Solutions to Linear Systems
PSPHS Particular Solution Plus Homogeneous Solutions
Section SS: Spanning Sets
SSNS Spanning Sets for Null Spaces
Section LI: Linear Independence
LIVHS Linearly Independent Vectors and Homogeneous Systems
LIVRN Linearly Independent Vectors, $r$ and $n$
MVSLD More Vectors than Size implies Linear Dependence
NMLIC Nonsingular Matrices have Linearly Independent Columns
NME2 Nonsingular Matrix Equivalences, Round 2
BNS Basis for Null Spaces
Section LDS: Linear Dependence and Spans
DLDS Dependency in Linearly Dependent Sets
BS Basis of a Span
Section O: Orthogonality
CRSM Conjugation Respects Vector Scalar Multiplication
IPVA Inner Product and Vector Addition
IPSM Inner Product and Scalar Multiplication
IPAC Inner Product is Anti-Commutative
IPN Inner Products and Norms
PIP Positive Inner Products
OSLI Orthogonal Sets are Linearly Independent
GSP Gram-Schmidt Procedure
Section MO: Matrix Operations
VSPM Vector Space Properties of Matrices
SMS Symmetric Matrices are Square
TMSM Transpose and Matrix Scalar Multiplication
TT Transpose of a Transpose
CRMSM Conjugation Respects Matrix Scalar Multiplication
CCM Conjugate of the Conjugate of a Matrix
MCT Matrix Conjugation and Transposes
AMSM Adjoint and Matrix Scalar Multiplication
Section MM: Matrix Multiplication
SLEMM Systems of Linear Equations as Matrix Multiplication
EMMVP Equal Matrices and Matrix-Vector Products
EMP Entries of Matrix Products
MMZM Matrix Multiplication and the Zero Matrix
MMIM Matrix Multiplication and Identity Matrix
MMDAA Matrix Multiplication Distributes Across Addition
MMSMM Matrix Multiplication and Scalar Matrix Multiplication
MMA Matrix Multiplication is Associative
MMIP Matrix Multiplication and Inner Products
MMCC Matrix Multiplication and Complex Conjugation
MMT Matrix Multiplication and Transposes
HMIP Hermitian Matrices and Inner Products
Section MISLE: Matrix Inverses and Systems of Linear Equations
TTMI Two-by-Two Matrix Inverse
CINM Computing the Inverse of a Nonsingular Matrix
MIU Matrix Inverse is Unique
SS Socks and Shoes
MIMI Matrix Inverse of a Matrix Inverse
MIT Matrix Inverse of a Transpose
MISM Matrix Inverse of a Scalar Multiple
Section MINM: Matrix Inverses and Nonsingular Matrices
NPNT Nonsingular Product has Nonsingular Terms
OSIS One-Sided Inverse is Sufficient
NI Nonsingularity is Invertibility
NME3 Nonsingular Matrix Equivalences, Round 3
SNCM Solution with Nonsingular Coefficient Matrix
UMI Unitary Matrices are Invertible
CUMOS Columns of Unitary Matrices are Orthonormal Sets
UMPIP Unitary Matrices Preserve Inner Products
Section CRS: Column and Row Spaces
CSCS Column Spaces and Consistent Systems
BCS Basis of the Column Space
CSNM Column Space of a Nonsingular Matrix
NME4 Nonsingular Matrix Equivalences, Round 4
REMRS Row-Equivalent Matrices have equal Row Spaces
BRS Basis for the Row Space
CSRST Column Space, Row Space, Transpose
Section FS: Four Subsets
PEEF Properties of Extended Echelon Form
FS Four Subsets
Section VS: Vector Spaces
ZVU Zero Vector is Unique
ZSSM Zero Scalar in Scalar Multiplication
ZVSM Zero Vector in Scalar Multiplication
AISM Additive Inverses from Scalar Multiplication
SMEZV Scalar Multiplication Equals the Zero Vector
Section S: Subspaces
TSS Testing Subsets for Subspaces
NSMS Null Space of a Matrix is a Subspace
SSS Span of a Set is a Subspace
CSMS Column Space of a Matrix is a Subspace
RSMS Row Space of a Matrix is a Subspace
LNSMS Left Null Space of a Matrix is a Subspace
Section LISS: Linear Independence and Spanning Sets
VRRB Vector Representation Relative to a Basis
Section B: Bases
SUVB Standard Unit Vectors are a Basis
CNMB Columns of Nonsingular Matrix are a Basis
NME5 Nonsingular Matrix Equivalences, Round 5
COB Coordinates and Orthonormal Bases
UMCOB Unitary Matrices Convert Orthonormal Bases
Section D: Dimension
SSLD Spanning Sets and Linear Dependence
BIS Bases have Identical Sizes
DCM Dimension of $\complex{m}$
DP Dimension of $P_n$
DM Dimension of $M_{mn}$
CRN Computing Rank and Nullity
RPNC Rank Plus Nullity is Columns
RNNM Rank and Nullity of a Nonsingular Matrix
NME6 Nonsingular Matrix Equivalences, Round 6
Section PD: Properties of Dimension
ELIS Extending Linearly Independent Sets
G Goldilocks
PSSD Proper Subspaces have Smaller Dimension
EDYES Equal Dimensions Yields Equal Subspaces
RMRT Rank of a Matrix is the Rank of the Transpose
DFS Dimensions of Four Subspaces
Section DM: Determinant of a Matrix
EMDRO Elementary Matrices Do Row Operations
EMN Elementary Matrices are Nonsingular
NMPEM Nonsingular Matrices are Products of Elementary Matrices
DMST Determinant of Matrices of Size Two
DT Determinant of the Transpose
Section PDM: Properties of Determinants of Matrices
DZRC Determinant with Zero Row or Column
DRCS Determinant for Row or Column Swap
DRCM Determinant for Row or Column Multiples
DERC Determinant with Equal Rows or Columns
DRCMA Determinant for Row or Column Multiples and Addition
DIM Determinant of the Identity Matrix
DEM Determinants of Elementary Matrices
DEMMM Determinants, Elementary Matrices, Matrix Multiplication
SMZD Singular Matrices have Zero Determinants
NME7 Nonsingular Matrix Equivalences, Round 7
DRMM Determinant Respects Matrix Multiplication
Section EE: Eigenvalues and Eigenvectors
EMHE Every Matrix Has an Eigenvalue
EMRCP Eigenvalues of a Matrix are Roots of Characteristic Polynomials
EMS Eigenspace for a Matrix is a Subspace
EMNS Eigenspace of a Matrix is a Null Space
Section PEE: Properties of Eigenvalues and Eigenvectors
EDELI Eigenvectors with Distinct Eigenvalues are Linearly Independent
SMZE Singular Matrices have Zero Eigenvalues
NME8 Nonsingular Matrix Equivalences, Round 8
ESMM Eigenvalues of a Scalar Multiple of a Matrix
EOMP Eigenvalues Of Matrix Powers
EPM Eigenvalues of the Polynomial of a Matrix
EIM Eigenvalues of the Inverse of a Matrix
ETM Eigenvalues of the Transpose of a Matrix
ERMCP Eigenvalues of Real Matrices come in Conjugate Pairs
DCP Degree of the Characteristic Polynomial
NEM Number of Eigenvalues of a Matrix
ME Multiplicities of an Eigenvalue
MNEM Maximum Number of Eigenvalues of a Matrix
HMRE Hermitian Matrices have Real Eigenvalues
HMOE Hermitian Matrices have Orthogonal Eigenvectors
Section SD: Similarity and Diagonalization
SER Similarity is an Equivalence Relation
SMEE Similar Matrices have Equal Eigenvalues
DC Diagonalization Characterization
DMFE Diagonalizable Matrices have Full Eigenspaces
DED Distinct Eigenvalues implies Diagonalizable
Section LT: Linear Transformations
LTTZZ Linear Transformations Take Zero to Zero
MBLT Matrices Build Linear Transformations
MLTCV Matrix of a Linear Transformation, Column Vectors
LTLC Linear Transformations and Linear Combinations
LTDB Linear Transformation Defined on a Basis
SLTLT Sum of Linear Transformations is a Linear Transformation
MLTLT Multiple of a Linear Transformation is a Linear Transformation
VSLT Vector Space of Linear Transformations
CLTLT Composition of Linear Transformations is a Linear Transformation
Section ILT: Injective Linear Transformations
KLTS Kernel of a Linear Transformation is a Subspace
KPI Kernel and Pre-Image
KILT Kernel of an Injective Linear Transformation
ILTLI Injective Linear Transformations and Linear Independence
ILTB Injective Linear Transformations and Bases
ILTD Injective Linear Transformations and Dimension
CILTI Composition of Injective Linear Transformations is Injective
Section SLT: Surjective Linear Transformations
RLTS Range of a Linear Transformation is a Subspace
RSLT Range of a Surjective Linear Transformation
SSRLT Spanning Set for Range of a Linear Transformation
RPI Range and Pre-Image
SLTB Surjective Linear Transformations and Bases
SLTD Surjective Linear Transformations and Dimension
CSLTS Composition of Surjective Linear Transformations is Surjective
Section IVLT: Invertible Linear Transformations
ILTLT Inverse of a Linear Transformation is a Linear Transformation
IILT Inverse of an Invertible Linear Transformation
ILTIS Invertible Linear Transformations are Injective and Surjective
CIVLT Composition of Invertible Linear Transformations
ICLT Inverse of a Composition of Linear Transformations
IVSED Isomorphic Vector Spaces have Equal Dimension
ROSLT Rank Of a Surjective Linear Transformation
NOILT Nullity Of an Injective Linear Transformation
RPNDD Rank Plus Nullity is Domain Dimension
Section VR: Vector Representations
VRLT Vector Representation is a Linear Transformation
VRI Vector Representation is Injective
VRS Vector Representation is Surjective
VRILT Vector Representation is an Invertible Linear Transformation
CFDVS Characterization of Finite Dimensional Vector Spaces
IFDVS Isomorphism of Finite Dimensional Vector Spaces
CLI Coordinatization and Linear Independence
CSS Coordinatization and Spanning Sets
Section MR: Matrix Representations
FTMR Fundamental Theorem of Matrix Representation
MRSLT Matrix Representation of a Sum of Linear Transformations
MRMLT Matrix Representation of a Multiple of a Linear Transformation
MRCLT Matrix Representation of a Composition of Linear Transformations
KNSI Kernel and Null Space Isomorphism
RCSI Range and Column Space Isomorphism
IMR Invertible Matrix Representations
IMILT Invertible Matrices, Invertible Linear Transformation
NME9 Nonsingular Matrix Equivalences, Round 9
Section CB: Change of Basis
CB Change-of-Basis
ICBM Inverse of Change-of-Basis Matrix
MRCB Matrix Representation and Change of Basis
SCB Similarity and Change of Basis
EER Eigenvalues, Eigenvectors, Representations
Section OD: Orthonormal Diagonalization
PTMT Product of Triangular Matrices is Triangular
ITMT Inverse of a Triangular Matrix is Triangular
UTMR Upper Triangular Matrix Representation
OBUTR Orthonormal Basis for Upper Triangular Representation
OD Orthonormal Diagonalization
OBNM Orthonormal Bases and Normal Matrices
Section CNO: Complex Number Operations
PCNA Properties of Complex Number Arithmetic
ZPCN Zero Product, Complex Numbers
ZPZT Zero Product, Zero Terms