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 | |

CRVA | Conjugation Respects Vector Addition |

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 |

TMA | Transpose and Matrix Addition |

TMSM | Transpose and Matrix Scalar Multiplication |

TT | Transpose of a Transpose |

CRMA | Conjugation Respects Matrix Addition |

CRMSM | Conjugation Respects Matrix Scalar Multiplication |

CCM | Conjugate of the Conjugate of a Matrix |

MCT | Matrix Conjugation and Transposes |

AMA | Adjoint and Matrix Addition |

AMSM | Adjoint and Matrix Scalar Multiplication |

AA | Adjoint of an Adjoint |

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 |

MMAD | Matrix Multiplication and Adjoints |

AIP | Adjoint and Inner Product |

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 |

AIU | Additive Inverses are 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 |

DER | Determinant Expansion about Rows |

DT | Determinant of the Transpose |

DEC | Determinant Expansion about Columns |

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 |

CCRA | Complex Conjugation Respects Addition |

CCRM | Complex Conjugation Respects Multiplication |

CCT | Complex Conjugation Twice |

Section SET: Sets |