Examples

Examples

 
Section WILA
Example TMP Trail Mix Packaging
 
Section SSLE
Example STNE Solving two (nonlinear) equations
Example NSE Notation for a system of equations
Example TTS Three typical systems
Example US Three equations, one solution
Example IS Three equations, infinitely many solutions
 
Section RREF
Example AM A matrix
Example NSLE Notation for systems of linear equations
Example AMAA Augmented matrix for Archetype A
Example TREM Two row-equivalent matrices
Example USR Three equations, one solution, reprised
Example RREF A matrix in reduced row-echelon form
Example NRREF A matrix not in reduced row-echelon form
Example SAB Solutions for Archetype B
Example SAA Solutions for Archetype A
Example SAE Solutions for Archetype E
 
Section TSS
Example RREFN Reduced row-echelon form notation
Example ISSI Describing infinite solution sets, Archetype I
Example FDV Free and dependent variables
Example CFV Counting free variables
Example OSGMD One solution gives many, Archetype D
 
Section HSE
Example AHSAC Archetype C as a homogeneous system
Example HUSAB Homogeneous, unique solution, Archetype B
Example HISAA Homogeneous, infinite solutions, Archetype A
Example HISAD Homogeneous, infinite solutions, Archetype D
Example NSEAI Null space elements of Archetype I
Example CNS1 Computing a null space, #1
Example CNS2 Computing a null space, #2
 
Section NM
Example S A singular matrix, Archetype A
Example NM A nonsingular matrix, Archetype B
Example IM An identity matrix
Example SRR Singular matrix, row-reduced
Example NSR Nonsingular matrix, row-reduced
Example NSS Null space of a singular matrix
Example NSNM Null space of a nonsingular matrix
 
Section VO
Example VESE Vector equality for a system of equations
Example VA Addition of two vectors in ${ℂ}^4$
Example CVSM Scalar multiplication in ${ℂ}^5$
 
Section LC
Example TLC Two linear combinations in ${ℂ}^6$
Example ABLC Archetype B as a linear combination
Example AALC Archetype A as a linear combination
Example VFSAD Vector form of solutions for Archetype D
Example VFS Vector form of solutions
Example VFSAI Vector form of solutions for Archetype I
Example VFSAL Vector form of solutions for Archetype L
Example PSHS Particular solutions, homogeneous solutions, Archetype D
 
Section SS
Example ABS A basic span
Example SCAA Span of the columns of Archetype A
Example SCAB Span of the columns of Archetype B
Example SSNS Spanning set of a null space
Example NSDS Null space directly as a span
Example SCAD Span of the columns of Archetype D
 
Section LI
Example LDS Linearly dependent set in ${ℂ}^5$
Example LIS Linearly independent set in ${ℂ}^5$
Example LIHS Linearly independent, homogeneous system
Example LDHS Linearly dependent, homogeneous system
Example LDRN Linearly dependent, $r<n$
Example LLDS Large linearly dependent set in ${ℂ}^4$
Example LDCAA Linearly dependent columns in Archetype A
Example LICAB Linearly independent columns in Archetype B
Example LINSB Linear independence of null space basis
Example NSLIL Null space spanned by linearly independent set, Archetype L
 
Section LDS
Example RSC5 Reducing a span in ${ℂ}^5$
Example COV Casting out vectors
Example RSC4 Reducing a span in ${ℂ}^4$
Example RES Reworking elements of a span
 
Section O
Example CSIP Computing some inner products
Example CNSV Computing the norm of some vectors
Example TOV Two orthogonal vectors
Example SUVOS Standard Unit Vectors are an Orthogonal Set
Example AOS An orthogonal set
Example GSTV Gram-Schmidt of three vectors
Example ONTV Orthonormal set, three vectors
Example ONFV Orthonormal set, four vectors
 
Section MO
Example MA Addition of two matrices in $M_{23}$
Example MSM Scalar multiplication in $M_{32}$
Example TM Transpose of a $3\times 4$ matrix
Example SYM A symmetric $5\times 5$ matrix
Example CCM Complex conjugate of a matrix
 
Section MM
Example MTV A matrix times a vector
Example MNSLE Matrix notation for systems of linear equations
Example MBC Money’s best cities
Example PTM Product of two matrices
Example MMNC Matrix multiplication is not commutative
Example PTMEE Product of two matrices, entry-by-entry
 
Section MISLE
Example SABMI Solutions to Archetype B with a matrix inverse
Example MWIAA A matrix without an inverse, Archetype A
Example MI Matrix inverse
Example CMI Computing a matrix inverse
Example CMIAB Computing a matrix inverse, Archetype B
 
Section MINM
Example UM3 Unitary matrix of size 3
Example UPM Unitary permutation matrix
Example OSMC Orthonormal set from matrix columns
 
Section CRS
Example CSMCS Column space of a matrix and consistent systems
Example MCSM Membership in the column space of a matrix
Example CSTW Column space, two ways
Example CSOCD Column space, original columns, Archetype D
Example CSAA Column space of Archetype A
Example CSAB Column space of Archetype B
Example RSAI Row space of Archetype I
Example RSREM Row spaces of two row-equivalent matrices
Example IAS Improving a span
Example CSROI Column space from row operations, Archetype I
 
Section FS
Example LNS Left null space
Example CSANS Column space as null space
Example SEEF Submatrices of extended echelon form
Example FS1 Four subsets, #1
Example FS2 Four subsets, #2
Example FSAG Four subsets, Archetype G
 
Section VS
Example VSCV The vector space ${ℂ}^m$
Example VSM The vector space of matrices, $M_{mn}$
Example VSP The vector space of polynomials, $P_n$
Example VSIS The vector space of infinite sequences
Example VSF The vector space of functions
Example VSS The singleton vector space
Example CVS The crazy vector space
Example PCVS Properties for the Crazy Vector Space
 
Section S
Example SC3 A subspace of ${ℂ}^3$
Example SP4 A subspace of $P_4$
Example NSC2Z A non-subspace in ${ℂ}^2$, zero vector
Example NSC2A A non-subspace in ${ℂ}^2$, additive closure
Example NSC2S A non-subspace in ${ℂ}^2$, scalar multiplication closure
Example RSNS Recasting a subspace as a null space
Example LCM A linear combination of matrices
Example SSP Span of a set of polynomials
Example SM32 A subspace of $M_{32}$
 
Section LISS
Example LIP4 Linear independence in $P_4$
Example LIM32 Linear independence in $M_{32}$
Example LIC Linearly independent set in the crazy vector space
Example SSP4 Spanning set in $P_4$
Example SSM22 Spanning set in $M_{22}$
Example SSC Spanning set in the crazy vector space
Example AVR A vector representation
 
Section B
Example BP Bases for $P_n$
Example BM A basis for the vector space of matrices
Example BSP4 A basis for a subspace of $P_4$
Example BSM22 A basis for a subspace of $M_{22}$
Example BC Basis for the crazy vector space
Example RSB Row space basis
Example RS Reducing a span
Example CABAK Columns as Basis, Archetype K
Example CROB4 Coordinatization relative to an orthonormal basis, ${ℂ}^4$
Example CROB3 Coordinatization relative to an orthonormal basis, ${ℂ}^3$
 
Section D
Example LDP4 Linearly dependent set in $P_4$
Example DSM22 Dimension of a subspace of $M_{22}$
Example DSP4 Dimension of a subspace of $P_4$
Example DC Dimension of the crazy vector space
Example VSPUD Vector space of polynomials with unbounded degree
Example RNM Rank and nullity of a matrix
Example RNSM Rank and nullity of a square matrix
 
Section PD
Example BPR Bases for $P_n$, reprised
Example BDM22 Basis by dimension in $M_{22}$
Example SVP4 Sets of vectors in $P_4$
Example RRTI Rank, rank of transpose, Archetype I
Example SDS Simple direct sum
 
Section DM
Example EMRO Elementary matrices and row operations
Example SS Some submatrices
Example D33M Determinant of a $3\times 3$ matrix
Example TCSD Two computations, same determinant
Example DUTM Determinant of an upper triangular matrix
 
Section PDM
Example DRO Determinant by row operations
Example ZNDAB Zero and nonzero determinant, Archetypes A and B
 
Section EE
Example SEE Some eigenvalues and eigenvectors
Example PM Polynomial of a matrix
Example CAEHW Computing an eigenvalue the hard way
Example CPMS3 Characteristic polynomial of a matrix, size 3
Example EMS3 Eigenvalues of a matrix, size 3
Example ESMS3 Eigenspaces of a matrix, size 3
Example EMMS4 Eigenvalue multiplicities, matrix of size 4
Example ESMS4 Eigenvalues, symmetric matrix of size 4
Example HMEM5 High multiplicity eigenvalues, matrix of size 5
Example CEMS6 Complex eigenvalues, matrix of size 6
Example DEMS5 Distinct eigenvalues, matrix of size 5
 
Section PEE
Example BDE Building desired eigenvalues
 
Section SD
Example SMS5 Similar matrices of size 5
Example SMS3 Similar matrices of size 3
Example EENS Equal eigenvalues, not similar
Example DAB Diagonalization of Archetype B
Example DMS3 Diagonalizing a matrix of size 3
Example NDMS4 A non-diagonalizable matrix of size 4
Example DEHD Distinct eigenvalues, hence diagonalizable
Example HPDM High power of a diagonalizable matrix
Example FSCF Fibonacci sequence, closed form
 
Section LT
Example ALT A linear transformation
Example NLT Not a linear transformation
Example LTPM Linear transformation, polynomials to matrices
Example LTPP Linear transformation, polynomials to polynomials
Example LTM Linear transformation from a matrix
Example MFLT Matrix from a linear transformation
Example MOLT Matrix of a linear transformation
Example LTDB1 Linear transformation defined on a basis
Example LTDB2 Linear transformation defined on a basis
Example LTDB3 Linear transformation defined on a basis
Example SPIAS Sample pre-images, Archetype S
Example STLT Sum of two linear transformations
Example SMLT Scalar multiple of a linear transformation
Example CTLT Composition of two linear transformations
 
Section ILT
Example NIAQ Not injective, Archetype Q
Example IAR Injective, Archetype R
Example IAV Injective, Archetype V
Example NKAO Nontrivial kernel, Archetype O
Example TKAP Trivial kernel, Archetype P
Example NIAQR Not injective, Archetype Q, revisited
Example NIAO Not injective, Archetype O
Example IAP Injective, Archetype P
Example NIDAU Not injective by dimension, Archetype U
 
Section SLT
Example NSAQ Not surjective, Archetype Q
Example SAR Surjective, Archetype R
Example SAV Surjective, Archetype V
Example RAO Range, Archetype O
Example FRAN Full range, Archetype N
Example NSAQR Not surjective, Archetype Q, revisited
Example NSAO Not surjective, Archetype O
Example SAN Surjective, Archetype N
Example BRLT A basis for the range of a linear transformation
Example NSDAT Not surjective by dimension, Archetype T
 
Section IVLT
Example AIVLT An invertible linear transformation
Example ANILT A non-invertible linear transformation
Example CIVLT Computing the Inverse of a Linear Transformations
Example IVSAV Isomorphic vector spaces, Archetype V
 
Section VR
Example VRC4 Vector representation in ${ℂ}^4$
Example VRP2 Vector representations in $P_2$
Example TIVS Two isomorphic vector spaces
Example CVSR Crazy vector space revealed
Example ASC A subspace characterized
Example MIVS Multiple isomorphic vector spaces
Example CP2 Coordinatizing in $P_2$
Example CM32 Coordinatization in $M_{32}$
 
Section MR
Example OLTTR One linear transformation, three representations
Example ALTMM A linear transformation as matrix multiplication
Example MPMR Matrix product of matrix representations
Example KVMR Kernel via matrix representation
Example RVMR Range via matrix representation
Example ILTVR Inverse of a linear transformation via a representation
 
Section CB
Example ELTBM Eigenvectors of linear transformation between matrices
Example ELTBP Eigenvectors of linear transformation between polynomials
Example CBP Change of basis with polynomials
Example CBCV Change of basis with column vectors
Example MRCM Matrix representations and change-of-basis matrices
Example MRBE Matrix representation with basis of eigenvectors
Example ELTT Eigenvectors of a linear transformation, twice
Example CELT Complex eigenvectors of a linear transformation
 
Section OD
Example ANM A normal matrix
 
Section NLT
Example NM64 Nilpotent matrix, size 6, index 4
Example NM62 Nilpotent matrix, size 6, index 2
Example JB4 Jordan block, size 4
Example NJB5 Nilpotent Jordan block, size 5
Example NM83 Nilpotent matrix, size 8, index 3
Example KPNLT Kernels of powers of a nilpotent linear transformation
Example CFNLT Canonical form for a nilpotent linear transformation
 
Section IS
Example TIS Two invariant subspaces
Example EIS Eigenspaces as invariant subspaces
Example ISJB Invariant subspaces and Jordan blocks
Example GE4 Generalized eigenspaces, dimension 4 domain
Example GE6 Generalized eigenspaces, dimension 6 domain
Example LTRGE Linear transformation restriction on generalized eigenspace
Example ISMR4 Invariant subspaces, matrix representation, dimension 4 domain
Example ISMR6 Invariant subspaces, matrix representation, dimension 6 domain
Example GENR6 Generalized eigenspaces and nilpotent restrictions, dimension 6 domain
 
Section JCF
Example JCF10 Jordan canonical form, size 10
 
Section CNO
Example ACN Arithmetic of complex numbers
Example CSCN Conjugate of some complex numbers
Example MSCN Modulus of some complex numbers
 
Section SET
Example SETM Set membership
Example SSET Subset
Example CS Cardinality and Size
Example SU Set union
Example SI Set intersection
Example SC Set complement
 
Section PT
 
Section F
Example IM11 Integers mod 11
Example VSIM5 Vector space over integers mod 5
Example SM2Z7 Symmetric matrices of size 2 over ${\mathbb{Z}}_7$
Example FF8 Finite field of size 8
 
Section T
 
Section HP
Example HP Hadamard Product
 
Section VM
Example VM4 Vandermonde matrix of size 4
 
Section PSM
 
Section ROD
Example ROD2 Rank one decomposition, size 2
Example ROD4 Rank one decomposition, size 4
 
Section TD
Example TD4 Triangular decomposition, size 4
Example TDSSE Triangular decomposition solves a system of equations
Example TDEE6 Triangular decomposition, entry by entry, size 6
 
Section SVD
 
Section SR
 
Section POD
 
Section CF
Example PTFP Polynomial through five points
 
Section SAS
Example SS6W Sharing a secret 6 ways