Section WILA: What is Linear Algebra? | |
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TMP | Trail Mix Packaging |

Section SSLE: Solving Systems of Linear Equations | |

STNE | Solving two (nonlinear) equations |

NSE | Notation for a system of equations |

TTS | Three typical systems |

US | Three equations, one solution |

IS | Three equations, infinitely many solutions |

Section RREF: Reduced Row-Echelon Form | |

AM | A matrix |

NSLE | Notation for systems of linear equations |

AMAA | Augmented matrix for Archetype A |

TREM | Two row-equivalent matrices |

USR | Three equations, one solution, reprised |

RREF | A matrix in reduced row-echelon form |

NRREF | A matrix not in reduced row-echelon form |

SAB | Solutions for Archetype B |

SAA | Solutions for Archetype A |

SAE | Solutions for Archetype E |

Section TSS: Types of Solution Sets | |

RREFN | Reduced row-echelon form notation |

ISSI | Describing infinite solution sets, Archetype I |

FDV | Free and dependent variables |

CFV | Counting free variables |

OSGMD | One solution gives many, Archetype D |

Section HSE: Homogeneous Systems of Equations | |

AHSAC | Archetype C as a homogeneous system |

HUSAB | Homogeneous, unique solution, Archetype B |

HISAA | Homogeneous, infinite solutions, Archetype A |

HISAD | Homogeneous, infinite solutions, Archetype D |

NSEAI | Null space elements of Archetype I |

CNS1 | Computing a null space, no. 1 |

CNS2 | Computing a null space, no. 2 |

Section NM: Nonsingular Matrices | |

S | A singular matrix, Archetype A |

NM | A nonsingular matrix, Archetype B |

IM | An identity matrix |

SRR | Singular matrix, row-reduced |

NSR | Nonsingular matrix, row-reduced |

NSS | Null space of a singular matrix |

NSNM | Null space of a nonsingular matrix |

Section VO: Vector Operations | |

VESE | Vector equality for a system of equations |

VA | Addition of two vectors in $\complex{4}$ |

CVSM | Scalar multiplication in $\complex{5}$ |

Section LC: Linear Combinations | |

TLC | Two linear combinations in $\complex{6}$ |

ABLC | Archetype B as a linear combination |

AALC | Archetype A as a linear combination |

VFSAD | Vector form of solutions for Archetype D |

VFS | Vector form of solutions |

VFSAI | Vector form of solutions for Archetype I |

VFSAL | Vector form of solutions for Archetype L |

PSHS | Particular solutions, homogeneous solutions, Archetype D |

Section SS: Spanning Sets | |

ABS | A basic span |

SCAA | Span of the columns of Archetype A |

SCAB | Span of the columns of Archetype B |

SSNS | Spanning set of a null space |

NSDS | Null space directly as a span |

SCAD | Span of the columns of Archetype D |

Section LI: Linear Independence | |

LDS | Linearly dependent set in $\complex{5}$ |

LIS | Linearly independent set in $\complex{5}$ |

LIHS | Linearly independent, homogeneous system |

LDHS | Linearly dependent, homogeneous system |

LDRN | Linearly dependent, $r$ and $n$ |

LLDS | Large linearly dependent set in $\complex{4}$ |

LDCAA | Linearly dependent columns in Archetype A |

LICAB | Linearly independent columns in Archetype B |

LINSB | Linear independence of null space basis |

NSLIL | Null space spanned by linearly independent set, Archetype L |

Section LDS: Linear Dependence and Spans | |

RSC5 | Reducing a span in $\complex{5}$ |

COV | Casting out vectors |

RSC4 | Reducing a span in $\complex{4}$ |

RES | Reworking elements of a span |

Section O: Orthogonality | |

CSIP | Computing some inner products |

CNSV | Computing the norm of some vectors |

TOV | Two orthogonal vectors |

SUVOS | Standard Unit Vectors are an Orthogonal Set |

AOS | An orthogonal set |

GSTV | Gram-Schmidt of three vectors |

ONTV | Orthonormal set, three vectors |

ONFV | Orthonormal set, four vectors |

Section MO: Matrix Operations | |

MA | Addition of two matrices in $M_{23}$ |

MSM | Scalar multiplication in $M_{32}$ |

TM | Transpose of a $3\times 4$ matrix |

SYM | A symmetric $5\times 5$ matrix |

CCM | Complex conjugate of a matrix |

Section MM: Matrix Multiplication | |

MTV | A matrix times a vector |

MNSLE | Matrix notation for systems of linear equations |

MBC | Money's best cities |

PTM | Product of two matrices |

MMNC | Matrix multiplication is not commutative |

PTMEE | Product of two matrices, entry-by-entry |

Section MISLE: Matrix Inverses and Systems of Linear Equations | |

SABMI | Solutions to Archetype B with a matrix inverse |

MWIAA | A matrix without an inverse, Archetype A |

MI | Matrix inverse |

CMI | Computing a matrix inverse |

CMIAB | Computing a matrix inverse, Archetype B |

Section MINM: Matrix Inverses and Nonsingular Matrices | |

UM3 | Unitary matrix of size 3 |

UPM | Unitary permutation matrix |

OSMC | Orthonormal set from matrix columns |

Section CRS: Column and Row Spaces | |

CSMCS | Column space of a matrix and consistent systems |

MCSM | Membership in the column space of a matrix |

CSTW | Column space, two ways |

CSOCD | Column space, original columns, Archetype D |

CSAA | Column space of Archetype A |

CSAB | Column space of Archetype B |

RSAI | Row space of Archetype I |

RSREM | Row spaces of two row-equivalent matrices |

IAS | Improving a span |

CSROI | Column space from row operations, Archetype I |

Section FS: Four Subsets | |

LNS | Left null space |

CSANS | Column space as null space |

SEEF | Submatrices of extended echelon form |

FS1 | Four subsets, no. 1 |

FS2 | Four subsets, no. 2 |

FSAG | Four subsets, Archetype G |

Section VS: Vector Spaces | |

VSCV | The vector space $\complex{m}$ |

VSM | The vector space of matrices, $M_{mn}$ |

VSP | The vector space of polynomials, $P_n$ |

VSIS | The vector space of infinite sequences |

VSF | The vector space of functions |

VSS | The singleton vector space |

CVS | The crazy vector space |

PCVS | Properties for the Crazy Vector Space |

Section S: Subspaces | |

SC3 | A subspace of $\complex{3}$ |

SP4 | A subspace of $P_4$ |

NSC2Z | A non-subspace in $\complex{2}$, zero vector |

NSC2A | A non-subspace in $\complex{2}$, additive closure |

NSC2S | A non-subspace in $\complex{2}$, scalar multiplication closure |

RSNS | Recasting a subspace as a null space |

LCM | A linear combination of matrices |

SSP | Span of a set of polynomials |

SM32 | A subspace of $M_{32}$ |

Section LISS: Linear Independence and Spanning Sets | |

LIP4 | Linear independence in $P_4$ |

LIM32 | Linear independence in $M_{32}$ |

LIC | Linearly independent set in the crazy vector space |

SSP4 | Spanning set in $P_4$ |

SSM22 | Spanning set in $M_{22}$ |

SSC | Spanning set in the crazy vector space |

AVR | A vector representation |

Section B: Bases | |

BP | Bases for $P_n$ |

BM | A basis for the vector space of matrices |

BSP4 | A basis for a subspace of $P_4$ |

BSM22 | A basis for a subspace of $M_{22}$ |

BC | Basis for the crazy vector space |

RSB | Row space basis |

RS | Reducing a span |

CABAK | Columns as Basis, Archetype K |

CROB4 | Coordinatization relative to an orthonormal basis, $\complex{4}$ |

CROB3 | Coordinatization relative to an orthonormal basis, $\complex{3}$ |

Section D: Dimension | |

LDP4 | Linearly dependent set in $P_4$ |

DSM22 | Dimension of a subspace of $M_{22}$ |

DSP4 | Dimension of a subspace of $P_4$ |

DC | Dimension of the crazy vector space |

VSPUD | Vector space of polynomials with unbounded degree |

RNM | Rank and nullity of a matrix |

RNSM | Rank and nullity of a square matrix |

Section PD: Properties of Dimension | |

BPR | Bases for $P_n$, reprised |

BDM22 | Basis by dimension in $M_{22}$ |

SVP4 | Sets of vectors in $P_4$ |

RRTI | Rank, rank of transpose, Archetype I |

Section DM: Determinant of a Matrix | |

EMRO | Elementary matrices and row operations |

SS | Some submatrices |

D33M | Determinant of a $3\times 3$ matrix |

TCSD | Two computations, same determinant |

DUTM | Determinant of an upper triangular matrix |

Section PDM: Properties of Determinants of Matrices | |

DRO | Determinant by row operations |

ZNDAB | Zero and nonzero determinant, Archetypes A and B |

Section EE: Eigenvalues and Eigenvectors | |

SEE | Some eigenvalues and eigenvectors |

PM | Polynomial of a matrix |

CAEHW | Computing an eigenvalue the hard way |

CPMS3 | Characteristic polynomial of a matrix, size 3 |

EMS3 | Eigenvalues of a matrix, size 3 |

ESMS3 | Eigenspaces of a matrix, size 3 |

EMMS4 | Eigenvalue multiplicities, matrix of size 4 |

ESMS4 | Eigenvalues, symmetric matrix of size 4 |

HMEM5 | High multiplicity eigenvalues, matrix of size 5 |

CEMS6 | Complex eigenvalues, matrix of size 6 |

DEMS5 | Distinct eigenvalues, matrix of size 5 |

Section PEE: Properties of Eigenvalues and Eigenvectors | |

BDE | Building desired eigenvalues |

Section SD: Similarity and Diagonalization | |

SMS5 | Similar matrices of size 5 |

SMS3 | Similar matrices of size 3 |

EENS | Equal eigenvalues, not similar |

DAB | Diagonalization of Archetype B |

DMS3 | Diagonalizing a matrix of size 3 |

NDMS4 | A non-diagonalizable matrix of size 4 |

DEHD | Distinct eigenvalues, hence diagonalizable |

HPDM | High power of a diagonalizable matrix |

FSCF | Fibonacci sequence, closed form |

Section LT: Linear Transformations | |

ALT | A linear transformation |

NLT | Not a linear transformation |

LTPM | Linear transformation, polynomials to matrices |

LTPP | Linear transformation, polynomials to polynomials |

LTM | Linear transformation from a matrix |

MFLT | Matrix from a linear transformation |

MOLT | Matrix of a linear transformation |

LTDB1 | Linear transformation defined on a basis |

LTDB2 | Linear transformation defined on a basis |

LTDB3 | Linear transformation defined on a basis |

SPIAS | Sample pre-images, Archetype S |

STLT | Sum of two linear transformations |

SMLT | Scalar multiple of a linear transformation |

CTLT | Composition of two linear transformations |

Section ILT: Injective Linear Transformations | |

NIAQ | Not injective, Archetype Q |

IAR | Injective, Archetype R |

IAV | Injective, Archetype V |

NKAO | Nontrivial kernel, Archetype O |

TKAP | Trivial kernel, Archetype P |

NIAQR | Not injective, Archetype Q, revisited |

NIAO | Not injective, Archetype O |

IAP | Injective, Archetype P |

NIDAU | Not injective by dimension, Archetype U |

Section SLT: Surjective Linear Transformations | |

NSAQ | Not surjective, Archetype Q |

SAR | Surjective, Archetype R |

SAV | Surjective, Archetype V |

RAO | Range, Archetype O |

FRAN | Full range, Archetype N |

NSAQR | Not surjective, Archetype Q, revisited |

NSAO | Not surjective, Archetype O |

SAN | Surjective, Archetype N |

BRLT | A basis for the range of a linear transformation |

NSDAT | Not surjective by dimension, Archetype T |

Section IVLT: Invertible Linear Transformations | |

AIVLT | An invertible linear transformation |

ANILT | A non-invertible linear transformation |

CIVLT | Computing the Inverse of a Linear Transformations |

IVSAV | Isomorphic vector spaces, Archetype V |

Section VR: Vector Representations | |

VRC4 | Vector representation in $\complex{4}$ |

VRP2 | Vector representations in $P_2$ |

TIVS | Two isomorphic vector spaces |

CVSR | Crazy vector space revealed |

ASC | A subspace characterized |

MIVS | Multiple isomorphic vector spaces |

CP2 | Coordinatizing in $P_2$ |

CM32 | Coordinatization in $M_{32}$ |

Section MR: Matrix Representations | |

OLTTR | One linear transformation, three representations |

ALTMM | A linear transformation as matrix multiplication |

MPMR | Matrix product of matrix representations |

KVMR | Kernel via matrix representation |

RVMR | Range via matrix representation |

ILTVR | Inverse of a linear transformation via a representation |

Section CB: Change of Basis | |

ELTBM | Eigenvectors of linear transformation between matrices |

ELTBP | Eigenvectors of linear transformation between polynomials |

CBP | Change of basis with polynomials |

CBCV | Change of basis with column vectors |

MRCM | Matrix representations and change-of-basis matrices |

MRBE | Matrix representation with basis of eigenvectors |

ELTT | Eigenvectors of a linear transformation, twice |

CELT | Complex eigenvectors of a linear transformation |

Section OD: Orthonormal Diagonalization | |

ANM | A normal matrix |

Section CNO: Complex Number Operations | |

ACN | Arithmetic of complex numbers |

CSCN | Conjugate of some complex numbers |

MSCN | Modulus of some complex numbers |

Section SET: Sets | |

SETM | Set membership |

SSET | Subset |

CS | Cardinality and Size |

SU | Set union |

SI | Set intersection |

SC | Set complement |