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2024-10-04
Summary of Changes
Early vs. Late Determinants
~~~~~~~~~~~~~~~~~~~~~~~~~~~
Changes to minimize reliance on determinants and characteristic
polynomials for existing material for eigenvalues and eigenvectors
* Intersection and sum of subspaces eventually necessary for
proving properties of the algebraic multiplicity of an eigenvalue,
especially theorem relating dimensions
New: Theorem SIIS, Definition SOS, Theorem SSIS, Theorem SID
* Expand Example CAEHW and de-emphasize the notion that it is
the "hard way" with a new title and acronym, now Example CASE.
* Formalize the basic connection between the singular matrix
A - lambda I and an eigenvalue lambda with new Theorem ESM.
Shortens proofs of Theorem ESM and Theorem EMRCP.
* Move Theorem MNEM earlier, with a new proof that does not
use the characteristic polynomial.
* Minor adjustment and enhancement of the phrasing of the
definition of similar matrices.
* Expand conclusion of Theorem SMEE and improve proof to
only emply characteristic polynomial for algebraic multiplicities.
* Simplify the proof of Theorem DED, without using
algebraic multiplicities.
* Move definitions and results for triangular matrices.
Definitions UTM and LTM move from Section OD to Section MO,
with new Example ULTM. Theorem NTM new in Section MO,
describes nonsingular triangular matrices, with Example NSTM
to illustrate. Move Theorem PTMT about the product of
triangular matrices moves from Section OD to Section MM.
Move Theorem ITMT about inverses of triangular matrices
from Section OD to Section MINM, and improve the proof.
New exercises: MO.M10, MO.T25, MM.T28, MINM.T50
* Convert superfluous Example BDE to Exercise PEE.M20.
v3.60 2024-10-04
~~~~~~~~~~~~~~~~
New: Exercise PEE.T21, eigenvalue proof two ways
New: Exercise VS.T31, alternate proof of Theorem AISM
New: Theorem SIIS, intersection of subspaces
New: Definition SOS, Theorem SSIS, sum of subspaces
New: Exercise S.M40, union of subspaces is not a subspace
New: Theorem SID, dimensions of intersection and sum of subspaces
New: Exercise PD.T60, alternate solution using Theorem SID
New: Theorem ESM, formalizing an eigenvalue discussion
New: Example ULTM, examples of triangular matrices
New: Theorem NTM, nonsingular triangular matrices
New: Example NSTM, nonsingular triangular matrices
New: Exercise MO.M10, both upper and lower triangular
New: Exercise MO.T25, transpose of triangular matrix
New: Exercise MM.T28, diagonal of triangular matrix product
New: Exercise MINM.T50, modified proof of Theorem NPNF
New: Example UE, unknown eigenvalues
New: Exercise MM.T80, apply nonsingular matrix to a linearly independent set
New: Exercise MM.T38, uniqueness of solutions with nonsingular coefficient matrices
New: Exercise O.T21, orthogonality in a special situation
Edit: Theorem TSS, added "an" in the proof (Rachel Chaiser)
Edit: Theorem EDYES, substantially shorten proof
Edit: Exercise IVLT.T15, streamline solution
Edit: Clarify Example NSE (Katrina Hoefflinger)
Edit: Added missing Sage practice cell in Sage LI (Samuel Gonzalez)
Edit: Theorem PSPHS, proof missing "+" (Katrina Hoefflinger)
Edit: Fine-tune induction proof of Theorem EOMP
Edit: Expanded Example MWIAA
Edit: Solution MR.T80, added one step to proof (Zach Jarvis)
Edit: Definition SUV, harmonize with Definition IM (Nathan Schultz)
Edit: Example SCAD, take more care with "span" (Robert Dwyer)
Edit: Structure five solutions as "iff" proofs (Mira Schoeberlein)
Edit: Simplify Solution ILT.T10
Edit: Definition CM, improve matrix display (Carissa Riedesel)
Edit: Clean-up Exercise VS.M13 (Jasmine Mikesell)
Edit: Expand Solution IVLT.T40
Edit: Improve Definition SIM, Similar Matrices
Edit: Theorem SMEE, expand conclusion, edit proof
Edit: Theorem DED, simplify the proof
Edit: Theorem DC, minor changes to streamline proof
Edit: Theorem SER, minor changes to streamline proof
Edit: Edit the proof of Theorem ITMT
Edit: Enhance discussion of nonsingular as similar to nonzero
Edit: Example EENS, take more care with conclusions drawn
Edit: Subsection SLT.SLTD, "are" changes to "is" (Jake McRae)
Change: Streamline second half of the proof of Theorem BS
Change: Exercise VO.T30, define subtraction more carefully (Ken Price)
Change: Theorems SSNS, BNS need case when n = r (Ken Price)
Change: Example NIDAU, in discussion, "are" to "is" (Lukas Karoly)
Change: Exercise LT.C43, clarify versions of the zero vector (Jake Rizzo)
Change: Solution SLT.C27, "are" to "is" (Ashlen Isbell)
Change: Rename Theorem NPNT as Theorem NPNF (Sam Webb)
Change: Streamline the proof of Theorem NPNF
Change: Expand Example CAEHW
Change: Re-title Example CAEHW to Example CASE
Change: Theorem EMRCP, shorter proof with new Theorem ESM
Change: Locate Theorem MNEM earlier, with new proof
Change: Locate Definition UTM and Definition LTM earlier
Change: Locate Theorem PTMT in Section MM
Change: Locate Theorem ITMT in Section MINM
Change: Reorganize proof of Theorem EDELI using induction
Change: Rename Theorem ZPZT as Theorem ZPZF
Change: Example BDE becomes Exercise PEE.M20
Sage: doctesting, version 9.2
Typo: Exercise SSLE.C53, add second condition to quoted problem (Jaime Ruiz)
Typo: Theorem PSPHS, stray addition sign in the proof (Rachel Chaiser)
Typo: Exercise MM.T12, correct reference, reword slightly (Matt Fergoda)
Typo: Theorem SMEZV, formatting error, second equation in Case 2 of proof (Matt Fergoda)
Typo: Exercises VS, group three cancellation exercises properly (Rachel Chaiser)
Typo: Solution EE.C19, correct two subscripts at the end (Matt Fergoda)
Typo: Theorem EIM, last line of proof need a zero vector (Matt Fergoda)
Typo: Exercise MR.T60, use correct linear transformation (Rachel Hirsch)
Typo: Section VR, "than" corrected to "then" (Rachel Chaiser)
Typo: Exercises ILT.C27, ILT.C28, and ILT.C29, "nullspace" is two words (Rachel Hirsch)
Typo: Section PD, "that" to "than" (Benjamin Millam)
Typo: Sage NS, "us" to "use" (Evan Eurs)
Typo: Sage SS3, missing "we" (Evan Eurs)
Typo: Example DRO, LaTeX macro for arrow incorrect (Patrick Zimmerman)
Typo: Index heading, "crazy vector space" (Samuel Gonzalez)
Typo: Theorem VFSLS, count non-pivot columns correctly (Rachel Chaiser)
Typo: Sage RCLS, add in a missing pronoun (Lukas Karoly)
Typo: Sage VSCV, "teh", extra "a" (Sam Keller)
Typo: Example TM, clean up unfinished edit (Ben Concannon)
Typo: Theorem BCS, missing period in statement (Jake Rizzo)
Typo: Theorem CUMOS, incorrect reason in proof (Uta Konishi)
Typo: Exercise LISS.C41, needed a space (Jake Rizzo)
Typo: Proof of Theorem CRN, "that that" (Bee Bridges)
Typo: Solution LISS.T51, Exercise ILT.T15, "that that"
Typo: Exercises EE.C23-C27, sets inside spans (Ashlen Isbell)
Typo: Exercise LISS.C42, sets inside spans
Typo: Solution ILT.C31, indexing on vectors (Jake Rizzo)
Typo: Solution MR.C25, missing period (Jake Rizzo)
Typo: Exercises MR.C50, MR.C51, coordinate linear transformation names (Ashlen Isbell)
Typo: Example SAE, correct column number for row operations (Jordan Calhoun)
Typo: Sage SSNS, "have have" (Jordan Calhoun)
Typo: Solution MO.C13, sign change (Marisa Nagata)
Typo: Sage CRS.C35, remove two stray "\quad" (Jordan Calhoun)
Typo: Definition EEF, correct notation for zero submatrix (Matt Garcia)
Typo: Example ILTVR, linear transformation definition missing "d" (Marisa Nagata)
Typo: Example CBCV, add "of" (Jordyn Probst)
Typo: Solution RREF.M40, add missing "I" (Jasmine Mikesell)
Edit: Theorem FVCS, prefer to have "r" be the number of pivot columns (Anthony Chapman)
Typo: Solution NM.M30, add closing parenthesis and move comma (Sophia Mcgough)
Typo: Sage SS, spelling "subsequent" (Kelsy Bryson, Maggie Hatt)
Typo: Example SCAD, "extra" D is the null space of D (Maggie Hatt)
Typo: Theorem BS, missing comma after ellipsis (Mira Schoeberlein)
Typo: Example COV, "the a" is just "the" (Maggie Hatt)
Typo: Sage RES, "commments" (Maggie Hatt)
Typo: Sage EVIC, misplaced question-mark for QQbar print representation (Nathan Schultz)
Typo: Property DSAM, add a space (Mira Schoeberlein)
Typo: Definition TM, add a space (Mira Schoeberlein)
Typo: Sections MO, CRS, write "matrix-vector" product (Carissa Riedesel)
Typo: Section MO, proper single-quote marks (Rowen Sanford-Eckhaus)
Typo: Property DSAC, add a space (Mira Schoeberlein)
Typo: Solution MM.T51, add a space (Sam Webb)
Typo: Ten missing spaces, each prior to inline math
Typo: Subsection MM.PMM, spell "multiplication" (Jasmine Mikesell)
Typo: Solution MO.C10, number parts of solution as in statement (Jasmine Mikesell)
Typo: Subsection MM.PMM, mark a scalar 1 as math (Mira Schoeberlein)
Typo: Subsection MISLE.CIM, correct references to previous matrices (Carissa Riedesel)
Typo: Solution MM.T52, comma after infinitive phrase (Sam Webb)
Typo: Example ESMS4, matrix is C, not A (Kelsy Bryson)
Typo: Exercise PEE.T21, alpha, not lambda
Typo: Exercise SD.C20, reference Exercise EE.C21, not EE.C20 (Mira Schoeberlein)
Typo: Example DMS3, nonsingular conditions improved with a comma (Maggie Hatt)
Typo: Solution D.M20, add necessary parentheses (Mira Schoeberlein)
Typo: Proof of Theorem SSS, add a space (Mira Schoeberlein)
Typo: Exercise SLT.M60, "a", not "an" (Maggie Hatt)
Typo: Proof of Theorem DEMMM, "proceeds", not "procede" (Maggie Hatt)
Typo: Subsection VS.EVS, "eight" definitions, not "six" (Carissa Riedesel)
Typo: Solution ILT.T10, "led", not "lead" (Sam Webb)
Typo: Proof of Theorem PEEF, cite Definition NM, not Definition NSM (Sophia McGough)
Typo: Solution VS.M15, enumerate parts of the solution (Jasmine Mikesell)
Typo: Solution LISS.C26, "null space", not "nullspace" (Jasmine Mikesell)
Typo: Section OD, Principal Axis Theorem ends with unitary matrix (Rachel Chaiser)
Typo: Two more "nullspace"
Typo: Preimage is spelled without a hyphen, approximately 70 times (Jasmine Mikesell)
Typo: Solution MR.T80, domain of composition is U (Jake Fischer)
Typo: Solution SSLE.C51, six-digit string needs an "f" (Dan Lovinger)
Typo: Theorem EMHE, leading coefficient should be "1" (Karl-Dieter Crisman)
Typo: Proof Technique D, minor capitalization typo (Karl-Dieter Crisman)
Typo: Fix subscripts in proof of Theorem DMFE
Typo: Sage PSHS, spell "semantics" correctly (Jasmine Mikesell)
Typo: Sage RRSM, B2 should be b2 (Nathan Schultz)
Typo: Sage GS, remove a stray comment (Chrissy Safranski)
Typo: Sage LNS, unbalanced parenthesis (Maggie Hatt)
Typo: Solution EE.M60, correct and clarify eigenvectors (Anna Van Boven)
Typo: Theorem SMEE, roles of A and B reversed (Jake Fischer)
Typo: Figure FTMR, discussion needs a "go" (Jake McRae)
v3.50 2015/12/30
~~~~~~~~~~~~~~~~
New: Exercise SSLE.C53, a 2x2 system from The Economist
New: Exercise O.T31, Gram-Schmidt twice (suggested by Chili Johnson)
New: Exercises MR.T40, MR.T41, vector space of linear transformations (Chili Johnson)
Change: Annotated Acronyms, Chapter R, removed three obsolete items
Change: Exercise PD.M50, deleted, now obsolete
Change: Remove matrix inverse Archetype commentary for rectangular matrices
Edit: Consistent titles on Sage nonsingular matrix series (Daniel York)
Edit: Examples CROB4, CROB3, had old-style inner products (Chris Thron)
Edit: Streamline half of proof of Theorem NPNT
Edit: Example FS1, clarified submatrices of extended echelon form (Chili Johnson)
Edit: Definition EEF, matrix M needed to be typeset as math (Jordan Fonseca)
Edit: Example FSAG, clarified two vectors are from spanning set (Chili Johnson)
Edit: Theorem SMEZV, additional justification in proof
Edit: Theorem SSLD, justifications in alternate proof (Chili Johnson)
Edit: Theorem AISM, added missing justification in proof (Jonathan Adams)
Edit: Improved justifications for eigenstuff proofs
Edit: Definition ELEM, fixed scalar multiple elementary matrix (Lukas Edman)
Edit: Clarify Solution ILT.C33 (Daniel York)
Edit: Consistent notation in Subsection VR (Daniel York)
Edit: Commas in matrix space notation (Kelsey Reed)
Edit: Clarified the infinite solution set in Example TMP (Dave Perkins)
Edit: Degree of the zero polynomial discussed in Solution S.T31 (Sam Burdick)
Edit: Guard against a trivial counterexample in Exercise IVLT.T30 (Christian Schulz)
Edit: Correct indices in proof of Theorem TMA (Marco Milletti)
Edit: Improvements to proof of Theorem EMMVP (Marco Milletti)
Edit: Explain why we rarely cite Theorem MMA (Marco Milletti)
Typo: Example NSS, sentence needs "which" to join clauses (Chili Johnson)
Typo: Example LLDS, set is R, not B (Daniel York)
Typo: Solution LDS.C52, set is T, not S (Molly Hazlehurst)
Typo: Annotated Acronyms, Chapter M, Theorem CSCS, question mark, missing "h" (Kelsey Reed)
Typo: Example DSM22, subspace is Z, not V (Kelsey Reed)
Typo: Exercise CRS.C20, "clolumn" should be column (Henry Woody)
Typo: Exercise SD.T15, identity matrices have size n, not 3 (Henry Woody)
Typo: Solution SD.T16, added necessary line break (Henry Woody)
Typo: Exercise LT.C42, codomain is complexes, not reals (Daniel York)
Typo: Sage LTS, codomain is C^4, not C^2 (Kelsey Reed)
Typo: Section IVLT, "to to" is just "to" (Molly Hazlehurst)
Typo: Exercise LT.M60, linear transformation is Z, not T (Kelsey Reed)
Typo: Subsection IVLT.RNLT, "rank of the codomain" is "dimension" (Chili Johnson)
Typo: Exercises M60 in ILT, SLT, IVLT, more T that are Z (Jonathan Adams)
Typo: Example KVMR, basis should not be written as a span (Chili Johnson)
Typo: Exercises ILT.C30, SLT.C26,C27,C28, spans need set brackets (Kelsey Reed)
Typo: Solution SD.C21, eigenspace computation missing zero eigenvalue (Troy Kakugawa)
Typo: Solution FS.C25, spans need set brackets (Troy Kakugawa)
Typo: Archetype L has rank 3, not rank 5
Typo: Exercise MO.M22, matrices transcribed incorrectly in solution (Tara Koskulitz)
Typo: Theorem MISM, order of matrix multiplication in proof reversed (Dave Perkins)
Typo: Exercise VO.T06, one justification in solution corrected (Tara Koskulitz)
Typo: Spelling correction (Karl-Dieter Crisman)
v3.40 2014/12/12
~~~~~~~~~~~~~~~~
Typo: Example LIP4, sign on 2\alpha_{3}x^2 term, twice (Albert Schueller)
Typo: Exercise ILT.C30, degree should be dimension (Deborah DeHovitz)
Typo: Exercise PD.T05, added in trivial hypothesis (Jason Preszler)
Change: Theorem CMVEI, proof incorrect about r < m (Mike Taper)
v3.30 2014/08/27
~~~~~~~~~~~~~~~~
New: Exercise OD.T30, proof for product of upper triangular matrices
Edit: Clarified part of Solution LISS.C25 (Riley Carpenter)
Edit: Grammar - many contractions and hyphens surveyed, possibly adjusted
Edit: Clarification in Proof Technique D (Voornaam "Marc" Achternaam)
Typo: Theorem SMS, Definition SYM, not SM (Nathan Laviste)
Typo: Five minor mistakes in Section FS (Sam Coates)
Typo: Example MCSM, "pivoot" (Trenten Tollen)
Typo: LaTeX/XML formatting, Solution TSS.T11 now in a knowl (Adam Frank)
Typo: Chapter SLE, seven typos (Adam Frank)
Typo: Section WILA, "studies" to "studied" (Balachandran S)
Typo: Example ISSI, reference should be Exercise TSS.C10 (MSK)
Typo: Sage ANC, "scalr", and nearby, corrected (Arnold Lebow)
Typo: Example NSR, replace original matrix by the correct one (Wes Garrett)
Change: Reworded Exercise NM.T12 so statement is true (Adam Frank)
v3.20 2013/12/28
~~~~~~~~~~~~~~~~
New: Exercise ILT.T10, solution added
New: Section CNO, Theorems ZPCN, ZPZT about zero divisors
New: Exercise DM.M30, row-equivalence and Theorem PEEF (suggested by Todd Pytel)
Change: Bijection defined, never used (Anna Dovzhik)
Edit: Clarified lead-in to Theorem PSSD (Michael DuBois)
Edit: Example CAEHW, added an "and" (Anna Dovzhik)
Edit: Example BDE, removed stale link (Anna Dovzhik)
Edit: Section VS, removed stale link to Section F
Edit: Minor edits in Sections VS, S, LISS, B, D, PD
Edit: Theorem CRN, reworded proof, correction included
Edit: Sage SM, took out unnecessary commas (Jenna Fontaine)
Edit: Section LT, split sentence before Theorem MOLT (Jenna Fontaine)
Edit: Section LT, rephrase sentence just prior to Example LTDB1 (Anna Dovzhik)
Edit: Rework lead-in sentence to Definition PI (Becky Hanscam)
Edit: Examples RAO, FRAN, parenthetical remark looks like function argument (Anna Dovzhik)
Edit: Proof of Theorem NME9, new commas for readability (Anna Dovzhik)
Typo: Sage B, "that" changes to "than" (Jenna Fontaine)
Typo: Theorem ELIS, "vector space" changed to "a vector space" (Jenna Fontaine)
Typo: Theorem SSS, proof, "also a vector space" (Jenna Fontaine)
Typo: LaTeX/XML formatting in Exercise VS.T10 (Chris Spalding)
Typo: Sage SUTH0, "row space of a matrix by the rows" (Jenna Fontaine)
Typo: Theorem CRN, proof, "Theorem BNS provides a basis" (Jenna Fontaine)
Typo: Theorem CRN, proof, "column without a leading 1" (Jenna Fontaine)
Typo: Subsection PD.GT, "the construction of a basis" (Jenna Fontaine)
Typo: Sage DMS, double "we" (Michael DuBois)
Typo: LaTeX/XML formatting in Solution LISS.C21 (Chris Spalding, Paul Makl)
Typo: Problem PD.T05, added missing "for" (Jenna Fontaine)
Typo: Exercise PD.T33, "set is basis" to "set is a basis" (Jenna Fontaine)
Typo: Exercise PD.T33, remove extra "to" (Jenna Fontaine)
Typo: Theorem SSLD, fixed bolding on vector subscript (Chris Spalding)
Typo: Theorem DRMM, space needed between two words (Anna Dovzhik)
Typo: Example CAEHW, "the" to "this" (Jenna Fontaine)
Typo: Sage CEVAL, "be" to "by" (Anna Dovzhik)
Typo: Theorem HMOE, second step of proof justified by Property DCN (Anna Dovzhik)
Typo: Proof DMFE, missing a comma (Jenna Fontaine)
Typo: Solution SD.C21, "is a diagonal matrix" (Jenna Fontaine)
Typo: Sage LTM, "little to do with" (Alyxandra Bock)
Typo: Exercises EE.C26, EE.C27, characteristic polynomials, lambda versus x (Anna Dovzhik)
Typo: Example LTPM, removed plural on "linear transformation" (Anna Dovzhik)
Typo: Sage LTS, removed extraneous "a" (Jenna Fontaine)
Typo: Sage LTM, "forgetten" becomes "forgotten" (Jenna Fontaine)
Typo: Sage LTB, removed extraneous "we" (Anna Dovzhik)
Typo: Sage LTB, added "y" onto "equality" (Anna Dovzhik)
Typo: Sage LTB, "to be" changed to "is" (Jenna Fontaine)
Typo: Sage LTB, changed "superfically" to "superficially" (Jenna Fontaine)
Typo: Theorem VSLT, added a comma (Jenna Fontaine)
Typo: Sage CILT, replaced percentage by words (Anna Dovzhik)
Typo: Sage CILT, "transformation" misspelled
Typo: Sage SLT, "likey" to "likely" (Jenna Fontaine)
Typo: Theorem SLTB, doubled "are" in proof (Jenna Fontaine)
Typo: Sage CSLT, changed "wil" to "will" (Jenna Fontaine)
Typo: Problem EE.C19 algebraic multiplicity of C, not A (Chris Spalding)
Typo: After Theorem ICLT, inserted "a" (Anna Dovzhik)
Typo: Sage CIVLT, "bijective" should be "surjective" (Anna Dovzhik)
Typo: After Diagram AIVS, "trhe" to "the" (Anna Dovzhik)
Typo: Solution IVLT.M30, matrices missing brackets (Jenna Fontaine)
Typo: Paragraph before Sage LTOE: "so" to "see" (Anna Dovzhik)
Typo: Solution IVLT.T30, make "functions" singular (Jenna Fontaine)
Typo: Sage VR, "providing" to "provide" (Anna Dovzhik)
Typo: Theorem IMILT, "an invertible matrix" (Becky Hanscam)
Typo: Example KVMR, period on penultimate sentence (Anna Dovzhik)
Typo: Proof of IMILT, capitalize "with" (Anna Dovzhik)
Typo: Exercise MR.C41, capitalize "this", delete a comma (Anna Dovzhik)
Typo: Exercise MR.C50, delete extra "a" (Anna Dovzhik)
Typo: Archetypes, kernel basis refers to "null space"
Typo: Sage CELT, period becomes question mark (Anna Dovzhik)
Typo: Sage CELT, two missing letters (Anna Dovzhik)
Typo: Section SSLE, "purse" to "pursue" (Anna Dovzhik)
Typo: Section VO, removed stray parenthesis in a vector (Anna Dovzhik)
v3.11 2013/10/21
~~~~~~~~~~~~~~~~
Edit: Minor edits in Sections WILA, SSLE, RREF, TSS, HSE, NM
Edit: Minor edits in Section VO, LC, SS, LI, LDS, O
Edit: Minor edits in Section MO, MM, MISLE, MINM, CRS, FS
Edit: Extended slightly the conclusion of Theorem HSC
Edit: Sage CNIP, removed obsolete discussion (Michael DuBois)
Edit: Stronger finish to proof of Theorem TT (Anna Dovzhik)
Edit: Wording in Exercise MM.T52 (Dan Drake)
Edit: Section VS, added a semi-colon (Gavin Tranter)
Edit: Theorem HMIP, more Theorem EMMVP motivation (Anna Dovzhik)
Change: Theorem ISRN demoted to Exercise TSS.T11
Change: Prefer "pivot columns" over "leading 1", Chapter SLE, V, M
Change: Diagram DTSLS, use Theorem CSRN rather than FVCS (Chris Beaulaurier)
Typo: C^n in statement of Theorem SLSLC
Typo: Section LI, "scenees" (Jenna Fontaine)
Typo: Section LI, "in linearly independent" (Becky Hanscam)
Typo: Example LDRN, use Theorem LIVRN as justification (Anna Dovzhik)
Typo: Sage RLD, "nest" to "next" (Gavin Tranter)
Typo: Theorem OSLI, "nozero" to "nonzero" (Anna Dovzhik)
Typo: Solution LDS.T55, B should be C at very end (Chris Spalding)
Typo: LaTeX/XML formatting in Solution LDS.T40 (Vladimir Yelkhimov)
Typo: Property ZM of Theorem VSPM, matrix, not vector (Michael DuBois)
Typo: LaTeX/XML formatting in Exercise MO.C10 (Anna Dovzhik, Dan Drake)
Typo: LaTeX/XML formatting in Theorem HMIP (Anna Dovzhik)
Typo: LaTeX/XML formatting in Theorem TTMI (Gavin Tranter)
Typo: Sage MISLE, "fuill" to "full" (Gavin Tranter)
Typo: Sage MO, "conjugage" to "conjugate" (Anna Dovzhik)
Typo: Theorem TTMI, "has an inverse" (Jenna Fontaine)
Typo: Sage CSCS, "coeffiicent" to "coefficient" (Jenna Fontaine)
Typo: Example CSTW, "columns of a matrix" (Jenna Fontaine)
Typo: Sage CSOC, "linearly idependent" to "linearly independent" (Jenna Fontaine)
Typo: LaTeX/XML formatting in Theorem REMRS (Jenna Fontaine)
Typo: Theorem BRS, two extra plus signs in proof (Jenna Fontaine)
Typo: Archetype B as homogeneous system fixed, Example HUSAB (Mike Daven)
Typo: Solution MM.C20, "-2" should be "-3" (Anna Dovzhik)
Typo: Proof PEEF, second paragraph, insert "product" (Anna Dovzhik)
Typo: Solution MO.T13 "1<=i<=n" to "1<=j<=n" (Anna Dovzhik)
Typo: Solution MISLE.C27 space in "at" (Jenna Fontaine)
Typo: Sage MISLE, nonexistent "CINMN" acronym to "CINM" (Bruno Mello)
Typo: Subsection VS.VS, grammar, agreement in number (Chris Spalding)
Typo: Solution VS.M12, total garbage from LaTeX/XML conversion (Jenna Fontaine)
Typo: Example SM32 "show" to "shown" (Jenna Fontaine)
Typo: Sage VS "thse" to "these" (Jenna Fontaine)
Typo: Solutions VS.C15, VS.C16, VS.C17, "be linear" to "be a linear" (Jenna Fontaine)
Typo: Example LIM32, stray addition symbol (Jenna Fontaine)
Typo: Section LISS, introduction, "linear" to "linearly" (Chris Spalding)
v3.10 2013/08/20
~~~~~~~~~~~~~~~~
New: Exercise MM.T12, Theorem HMIP reprised
Change: Proof of Theorem OD uses normality of diagonal matrices (Hunter Wills)
Typo: Miscellaneous edits (David Farmer)
Typo: Solution ILT.C29, inner span should be a set (Aysha Orta)
Typo: Misspelling in Acknowlegements (Peter Wimberger)
v3.00 2012/12/05
~~~~~~~~~~~~~~~~
New: Diagrams NSLT, SLT (surjective linear transformations)
New: Diagrams KPI, IVLT (pre-images, invertible linear transformations)
New: Diagram KRI (kernel and range isomorphisms)
New: Exercise NM.T31 (with solution)
New: Exercises LT.T30 and LT.T31
New: Exercise LISS.T51, converse of Theorem VRRB
New: Exercise LISS.T20, subset of linearly independent set is linearly independent
New: Reading Questions, Section OD
New: Diagram AIVS, Section IVLT
New: Discussion following Theorem OBNM
New: Exercises MM.T35, OD.T10, OD.T20
New: Exercise VR.M20 (Tyler Ueltschi)
Edit: Explained min(m,n) in Exercise PD.T15
Edit: More on induction in Proof Technique I (Dan Messenger)
Edit: Added multiplicities to Solution EE.C19 (Duncan Bennett)
Edit: Subsection IVLT.SLELT, reference Definition MVP for T's definition (Kyle Whitcomb)
Edit: Completed Solution ILT.C27 (Duncan Bennett)
Edit: About 130 line-break adjustments, for 5 inch text width at 10pt
Edit: Edits, Subsection CB.CELT
Edit: Solution SLT.C29, added solution to surjective question (Dan Messenger)
Edit: Removed two forward references to Chapter MD (Aidam Meacham)
Edit: Proof of Theorem NME7 clarified (Kevin Halasz)
Edit: Rewrote discussion between Theorem EIM and Theorem ETM (Kevin Halasz)
Change: Replaced Definition RR by an informal discussion
Change: Definition TSVS renamed as Definition SSVS
Change: Proof Technique D: discussion about definitions as nouns, adjectives
Change: Subsection PD.DS (Direct Sums) removed, now in second volume
Change: Subsection MM.PMM, before Theorem MMIP, conjugate fixed to reflect theorem (Dave Perkins)
Change: Solution SLT.C25 no longer uses rank of a linear transformation (Becca Ebert)
Change: Proof of Theorem OD rewritten
Change: Theorem UTMR, proof changed to remove dependence on direct sums (Tyler Ueltschi)
Typo: Proof of Theorem SLSLC: Doubled text macro in source (Glenna Toomey)
Typo: Solution S.T30: Corrected bolding on 'Proof' (Glenna Toomey)
Typo: LaTeX/XML formatting in Solutions RREF.C11, RREF.C13, RREF.C16 (Caden Robinson)
Typo: LaTeX/XML formatting in Solutions SSLE.M10, LISS.C21, B.C11 (Caden Robinson)
Typo: Exercise TSS.T20, fixed link to notation (Caden Robinson)
Typo: Spelling and grammar, Annotated Acronyms VS (Nick Butterfield)
Typo: LaTeX/XML formatting, Proof of SLSLC, discussion after VSPCV
Typo: Several LS(A,b) without bold b vector (Sophie Bartolomeo)
Typo: Sign change deep in Example SSP (Dan Messenger)
Typo: LaTeX/XML formatting in Exercises MISLE.C41, MINM.C40 (Caden Robinson)
Typo: LaTeX/XML formatting in Exercise VS.M12 (Ameneh McCullough)
Typo: Removed bad forward reference in Exercise LISS.C21 (Tyler Ueltschi)
Typo: LaTeX/XML formatting in Exercises LISS.T50, D.M20 (Becca Ebert)
Typo: Grammar in proof of Theorem TSS (Frankie de Lavis)
Typo: Spelling in Preface "eigenvvectors"
Typo: LaTeX/XML formatting in Exercise PD.C40 (Becca Ebert)
Typo: Stray percent sign in Example D33M (Glenna Toomey)
Typo: Incorrect initial matrix in Solution DM.C30 (Julie Nelson)
Typo: Spelling in introduction to Section EE "ethe" (Liz Meuci)
Typo: Spelling in Section VS "be we" to "but we" (Caden Robinson)
Typo: Reference in Example VSM is to a definition, not example (Caden Robinson)
Typo: LaTeX/XML formatting in Reading Question LISS.2 (Caden Robinson)
Typo: LaTeX/XML formatting in Solutions EE.C23, EE.C24, EE.C25 (Caden Robinson)
Typo: LaTeX/XML formatting in Example DAB (Ben Boe)
Typo: Exercise PEE.T22 "had" to "has" (Dan Messenger)
Typo: Example SAV, misleading misalignment in matrix (Mitch Benning, Davis Schubert)
Typo: LaTeX/XML formatting in Exercise, Solution RREF.M40 (Aidan Meacham)
Typo: LaTeX/XML formatting in Solution ILT.C40 (Julie Nelson)
Typo: Section LT, after Example MFLT, "not an accident" (Kyle Whitcomb)
Typo: Section SLT, before Diagram SLT, domain/codomain confusion (Kyle Whitcomb)
Typo: Section IVLT, after Definition IVS, "given", "a" reversed (Kyle Whitcomb)
Typo: LaTeX rules for fill-in-the-blank exercises (Julie Nelson)
Typo: LaTeX/XML formatting in Solution IVLT.C20 (Julie Nelson)
Typo: LaTeX/XML formatting in Solution LT.C43 (Duncan Bennett)
Typo: Example SEEF, mis-aligned "M = " (Dave Perkins)
Typo: Example SPIAS, scalar in C^3 (Nick Butterfield)
Typo: LaTeX/XML formatting, \matrixrep in Theorem OBUTR proof
Typo: LaTeX/XML formatting, \lt in Theorem VRI proof (Stephen Gower)
Typo: Implication arrow in wrong direction, Theorem IMR (Dan Messenger)
Typo: Example LIC, second sentence, in/dependence (Aidan Meacham)
Typo: Solution B.T50, missing implication arrows
Typo: Sage CELT, needed symbolic variables (Tyler Ueltschi)
Typo: Solution CB.C40, uses B in solution; problem now asks for C (Tyler Ueltschi)
Typo: Missing reference to Exercise OD.T20 (Tyler Ueltschi)
Typo: Exercise MR.C25, added "is" (Caden Robinson)
Typo: Exercise VR.C20, final matrix has entry 40, not 30 (Melissa Pellman)
Typo: Example RVMR, variable change in linear transformation definition (Glenna Toomey)
Typo: Proof of Theorem SMZE, now cites Definition NM, not NSM (Kevin Halasz)
MAJOR SOURCE FORMAT CHANGE
~~~~~~~~~~~~~~~~~~~~~~~~~~
Version 2.99 is the last version that is guaranteed to compile from the
LaTeX sources initiated in December 2003. Content has all migrated
to an XML format, using a one-off conversion process, custom XML tags
and LaTeX syntax for in-line and displayed mathematics.
Changes to the LaTeX sources subsequent to Version 2.99 are
designed to support the automated conversion to XML and some
new environments (e.g. "introduction", "para", "graphics") are
not implemented in the corresponding style files. Additional
content has been integrated, such as Tikz graphics and the Sage
vignettes.
v2.99 source (as published, or with additions) may be recreated
in two ways:
(a) rolling back to the version "2.99" Mercurial tag (published version)
(b) in the new "latex" subdirectory (source with additions).
v2.99 2012/10/13
~~~~~~~~~~~~~
Omission: Corrected discussion of inner product definition (Diego Barreiro)
Typo: Set S is really T, discussion following Theorem BCS (Diego Barreiro)
Typo: Miscellaneous small corrections, Sections MM, MISLE, VS, S (Diego Barreiro)
Typo: Solution SSLE.M70, spelling (Becca Ebert)
Typo: Small fix to new version of VO.T05 (Nick Butterfield)
Typo: Solution LI.M50, "we that" becomes "we find that" (Dan Messenger)
Typo: Several "the the" (Jack Gilbert)
Edit: Theorem SLSLC statement: clarified that the vector x has n entries (Jack Gilbert)
Change: Alignment of several exercises in Section RREF (Ameneh McCullough)
Change: Solutions EE.C23, EE.C24 reformatted
Change: Exercises VO.T05, VO.T06, VO.T07: proofs re-written
v2.90 2012/06/17
~~~~~~~~~~~~~
MAJOR CHANGE:
The inner product is now "reversed" from what it was formerly.
Specifically, the entries of the *first* vector are conjugated now.
Rationale:
This will make properties for numerous matrix decompositions and other
aspects of orthogonality simpler and more natural. This is already
evident as several proofs have been shortened and simplified.
Changes:
Definition IP - root cause
Theorem IPSM - now scalar multiples of first vector
"come out" conjugated
Theorem MMIP - conjugate and transpose both now affect first vector
Theorem GSP - order in inner products reversed. Necessary for the proof
but also preserves any existing computations
Theorem COB - order in inner products reversed. Necessary for the proof
but also preserves any existing computations
Proofs - Many proofs depend on the five items above and have been
rewritten. Generally they are equivalent or simpler. These
changes are not documented here. See the revision history, which
is organized by section, or search on uses of the theorems above.
Limitations - Changes only go through Section CB. Remaining sections will
be adjusted as edited.
New: Exercise HSE.T11, equivalent systems, both homogeneous, or both not (Robert Beezer)
New: Exercise SSLE.M14, 'time flies like...' (Gary Gordon)
Typo: Extra "a" in Section VO (Aaron Taylor)
Typo: a/the Example LINSB (Zeke Rosenberg)
Typo: "see" missing from Solution LDS.C50 (Aaron Taylor)
Typo: "of" missing from Subsection MISLE.MI (Zeke Rosenberg)
Typo: Exercise and Solution SS.M12 needs set brackets in spans (Cliff Berger)
Typo: Proof of Theorem NI has bad reference to Theorem NMRRI (Troy Cornelius)
Typo: Adjusted subsection acronym to FS.CCS (Troy Cornelius)
Typo: Solution MO.C13, alpha = 2 (Preston Van Buren)
Typo: Exercises MO.C12, MO.C13 are matrix equations (Preston Van Buren)
Typo: Subsection LISS.LI, bad reference to column vector linear independence (Troy Cornelius)
Typo: Solution MINM.M15, nonsingular should be singular (Preston Van Buren)
Typo: Exercise MINM.M10, missing comma (Sophie Scheller)
Typo: Theorem DNLT, missing "if" (Shepherd Engle)
Typo: Solution LISS.C26, five matrices span, not four (Alden Horowitz)
Typo: Two minor ones, LISS reading questions (Sophie Scheller, Preston Van Buren)
Typo: Subsection EE.EEE, bad reference to determinant definition (David Farmer)
Typo: Example KVMR, linear transformation definition fixed (Ryan Walch)
Typo: Example CIVLT, kernel should be trivial (Ryan Walch, Cliff Berger)
Typo: Solution LISS.C22, dro is for (Cliff Berger)
Typo: Section MINM, The is This (Aaron Taylor)
Typo: Section MINM, calls is call (Troy Cornelius)
Typo: Example VRP2, second basis is C (Preston Van Buren)
Typo: Subsection NLT.CFNLT, dangling reference deleted (Shepherd Engle)
Typo: Exercise VS.M21, vector multiplication is scalar multiplication (Davis Shurbert)
Typo: Subsection MR.NRFO, reference to Theorem ME is Theorem EMP (Cliff Berger)
Typo: Solution S.C17, matrix needs augmenting (Alden Horowitz)
Typo: Solution SSLE.C51, removed asterisk as multiplication
Typo: Solution PD.C40, vector space is P_4, not P_5 (Cliff Berger)
Typo: Exercise ILT.T15, kernels, not null spaces
Typo: Solution CB.C41, S_{22} is domain (Preston Van Buren)
Typo: Archetypes, inverse linear transformation check reordered (Preston Van Buren)
Addition: Solution IVLT.T15 cites Definition ROLT (Alex King)
Change: Reworded Exercise TSS.M45 (Davis Shurbert)
Change: Subsection HP.DMHP, converted citation request (Troy Cornelius)
Change: Theorem SSS, clarified span is a subset, fixed missing brackets (Luran Li)
Change: Exercise NM.T10, Solution NM.T10 slightly reworked
Change: Exercises VO, reformatted 3 problem numbers
Change: Exercise VS.M12, Exercise VS.M13, changed addition symbol
Change: Reading Question MISLE.1, totally new matrix, with fraction-free inverse
v2.30 2011/12/23
~~~~~~~~~~~~~
Typo: Bad subscript in proof of Theorem RREFU (Antti Tuppurainen)
Typo: Misspellings in Solution LISS.C26 (Robert Tunney)
Typo: Misspellings in Annotated Acronyms VS (Robert Tunney)
v2.23 2011/11/30
~~~~~~~~~~~~~
Typo: Solution DM.M16, one root is -4 (Tuo "Thomas" Huang)
Typo: Solution LI.M21, alpha_3 subscripts fixed (Robert Tunney)
Typo: Exercise MO.C10, sign change on beta (Jessica Norworth, Robert Tunney)
Typo: Exercise SLT.C24, removed non-existent coefficient e (Audrey Tiew)
Typo: Example SSP4, two instances of 24 alpha_2 changed to 24 alpha_4
Typo: Section LC: "will be appear again" fixed (David Randolph)
New: Exercise VS.T30, Property O is just for 1 (James Gallagher)
New: Exercise EE.T15, two definitions of characteristic polynomial
New: Exercises M60 in LT, ILT, SLT, IVLT, MR about the zero linear transformation
New: Exercise IVLT.T40, composition invertible, then so are parts
New: Exercises MR.M80,M81,M82,M83: old matrix results in linear transformation terms
Change: Reading Question LDS.1
Change: Exercise SSLE.M70 fixed to work over complexes for one and zero solutions (Don Kreher)
Change: Improve discussion following Theorem EMMVP (Robert Tunney)
Change: Improve Solution SSLE.M10 (Robert Tunney)
Change: Exercise LI.C22, totally replaced to make linear combination simpler computationally (Robert Tunney)
Change: Improve wording of Definition EEF (Zoltan Toth)
Change: Careful reductions to echelon form in RREF run on index j (Ben Zeilstra)
v2.22 2010/04/21
~~~~~~~~~~~~~
Typo: Section LT intro, "He" -> "Here" (Sigrun Bodine)
Typo: Section LT, "how how" (Jinshil Yi)
Typo: Section EE, "take the each" (Jinshil Yi)
Typo: Theorem DEM, third part of proof, confused subscripts (Jinshil Yi)
Typo: Subsection ILT.ILTLI, "if" -> "in" (Shepherd Engle)
New: Expanded intro to Section MISLE
New: Expanded Solution MISLE.T10
New: Added Theorem NPNT to Annotated Acronyms in Chapter M
New: Added proper citations in first half of proof of Theorem TSS
New: Added m <= n to hypotheses of Theorem DSFB (Jinshil Yi)
Change: Exercise LISS.T40, removed references to a basis, reworded (Jinshil Yi)
Change: First half of proof of Theorem NI shortened
Removed: Exercises MM.M30, MM.M31. Duplicated in MM.T41.
v2.21 2010/03/10
~~~~~~~~~~~~~
Typo: Solution ILT.T20, null spaces of linear transformations should be kernels (Scott Macha, Lisa Kant)
Typo: Technique DC, "anitderivative" (Jinshil Yi)
Typo: "taht", "scalked" (Jinshil Yi)
Typo: Section VO, last paragraph, Property ACC corrected to AAC (David Eva)
Edit: Part 1 of Definition RREF slightly reworded.
Edit: Row operation examples and solutions reformatted
Edit: Improved \setparts macro
Edit: Various clean-up in Chapters SLE, V
New: Exercise HSE.T12 (Ivan Kessler)
New: Exercise NM.T12
v2.20 2009/08/17
~~~~~~~~~~~~~
Typo: Theorem GESD, missing power of m three times in kernels at very end. (Zoltan Toth)
Typo: Section JCF, Subsection JCF, introductory paragraph, two words missing. (Zoltan Toth)
Typo: Example DSM22, Coefficient of d, second equation, changed to -2. (Joe Gain)
Typo: Theorem DERC, "rows r and s" changed to "rows s and t." (Charles Ganiear)
Typo: Exercise SLT.C23, x^4 changed to x^3. (Tanya Leise)
Typo: Solution MR.C40, stray S changed to R. (Tanya Leise)
Typo: Exercise IVLT.C25, codomain to C^3 in part 5. (Tanya Leise)
Notation: Linear transformation definitions now use \rightarrow, not \mapsto. (Tanya Leise)
Change: Solutions RREF.C10-C19 rewritten to not reference next section (Manley Perkel)
Add: Example RREF.RREF has additional illustration of sets D and F. (Manley Perkel)
Add: Solution RREF.T10 has some extra explanation. (Robert Beezer)
New: Definitions SSLE and SSSLE in Section SSLE making "solution" a precise term
New: Exercises TSS.M46, TSS.M70, TSS.T20 (r, D, F notation) (Manley Perkel)
New: Exercise LI.M51 (Manley Perkel, Rob Beezer)
New: Exercises LI.M20, LI.M21 (Rob Beezer)
New: Edition formatted for Kindle DX
v2.12 2009/04/08
~~~~~~~~~~~~~
Typo: Proof Technique I, fixed algebra in proof for sum of integers (William Haynes)
Typo: Example CFV, for Archetype E, leading 1 is now row 3, column 5 (William Haynes)
Typo: Theorem CFNLT, table at end and following. Subscripts d corrected to p. (Zoltan Toth)
Typo: Theorem GESD, 2 typos, and a missing identity matrix at end. (Zoltan Toth)
Typo: Theorem GESD, T:V->V in hypothesis.
Typo: Solution EE.C24, wandering minus sign in eigenvector for eigenvalue 3. (Hung An Kim)
Typo: Removed ten stray instances of exam "points." (Tanya Leise)
Change: Theorem HMIP, second half of proof streamlined (Manley Perkel)
Change: Example VSF generalized slightly (Manley Perkel)
New: Enhancements to Definition VS and associated commentary (Manley Perkel)
New: Exercise O.M60 (Manley Perkel)
New: Exercises MO.T30--MO.T37, sequence of proofs about skew-symmetric matrices (Manley Perkel)
v2.11 2009/02/18
~~~~~~~~~~~~~
New: Links in problem statements to solutions vanish if solutions are omitted (Nancy Neudauer)
New: Edition file for Pacific University course (Nancy Neudauer)
New: Warning after Definition RREF abut the use of sets D and F for any matrix (Nancy Neudauer)
New: Exercises LI.C20-25 now ask for relation of linear dependence for linearly dependent sets (Manley Perkel)
New: Exercises VO.T30,T31,T32 about vector subtraction
New: Comments at the end of Section VO about subtraction, parentheses (Manley Perkel)
New: Partial fraction example added to Proof Technique DC as an example (Manley Perkel)
Change: Solution VS.M21 duplicates VS.M20, replaced with accurate solution (Tanya Leise)
Change: Definition A (adjoint) shouldn't require the matrix to be square (Manley Perkel)
Change: Example SCAB no longer mentions V, it was not used (Manley Perkel)
Change: Complex numbers in Subsection CNO.CCN written as Greek letters (Manley Perkel)
Change: Various grammar, spelling and other edits (Manley Perkel)
Typo: Definition F, four V's corrected to be F's (Tomas da Silva Martins de Godoi)
Typo: Theorem SLSLC, four 1 <= i <= n replaced with 1 <= i <= m (Valeron JC)
Typo: Solution RREF.M50 now mentions the number of trucks in the solution (Manley Perkel)
v2.10 2009/01/11
~~~~~~~~~~~~~
Typo: Archetype R, inverse linear transformation, 221/4 corrected to 221/2 (David Braithwaite)
Typo: Theorem CFNLT, third paragraph. Two s_i corrected to s_p (Zoltan Toth)
Typo: Theorem GESD, "scalarlistpn" macro fixed (Zoltan Toth)
Typo: Theorem GESD, missing "+" inserted deep into proof (Zoltan Toth)
Change: Theorem SS, reordered conclusions (Zoltan Toth)
Change: Theorem GESIS, new argument for invariance (Zoltan Toth)
New: Exercises from Chris Black, 123 in total, almost all with solutions:
SSLE: C30,C31,C32,C33,C34
RREF: M45
TSS: C21,C22,C23,C24,C25,C26,C27,C28
HSE: C21,C22,C23,C25,C26,C27
VO: C11,C12,C13,C14,C15,T5,T6,T7
SS: M10,M11,M12
MO: C10,C11,C12,C13,C14
MM: C21,C22,C23,C24,C25,C26,C30,C31,C32,C33,M30,M31
MISLE: C16,C17,C18,C19
MINM: C20,M10,M11,M15
CRS: C20
VS: M11,M12,M13,M14,M15,M20,M21
S: C15,C16,C17,T30,T31
LISS: C24,C25,C26
B: C10,C11,C12,C13,C14
D: C21,C22,C23,C35,C36,C37
DM: C21,C22,C23,C27,C28,C29,C30,M10,N11,M15,M16
EE: C10,C11,C12,C23,C24,C25,C26,C27
LT: C16,C40,C41,C42,C43
ILT: C26,C27,C28,C29,C30,C31,C32,C33
SLT: C23,C24,C26,C27,C28,C29,C30
IVLT: C25
v2.02 2008/11/19
~~~~~~~~~~~~~
Typo: Proof of Theorem RREFU, fixed [R]_{k,d_l} subscript (Niall Davidson)
Typo: Example CNS2 (Section HSE), augmented matrix needs column of zeros (Cicero Vargas)
Typo: Subsection FS.EEF, very end, augmented matrix [A|x] corrected to [A|y] (David Braithwaite)
Typo: Proof of Theorem UMCOB, inner product equals 0 (not 1) in beginning (Bryan Smith)
Typos: 117 typos and minor edits (Minh Van Nguyen)
Edits: Various edits and typos (Bryan Smith)
Edit: Theorem ETM, removed unnecessary eigenvector (David Braithwaite)
Edit: Theorem DEC, i/j subscript fixed in proof
Change: Solution MM.T41 rewritten, no longer depends on Theorem NI (David Braithwaite)
Change: Theorem DT, induction step made explicit (David Braithwaite)
v2.01 2008/09/10
~~~~~~~~~~~~~
New: Another solution for Exercise SSLE.C52(David Braithwaite)
New: Computation Notes for SAGE: RR (Row Reduce), LS (Linear Solve), VLC (Vector Linear Combinations)
Change: Theorem VFSLS, bad subscripts in paragraph "Rearranging.." (David Braithwaite)
Change: Theorem VFSLS, proof totally rewritten, includes correction above
Change: Exercise SSLE.C51 clarified, new solution (Arnim Littek)
Edits: Minor clean-up on SAGE Computation Notes
Typo: Solution NM.C50, B renamed properly to E (George Garrett)
Typo: Example NSDS (Section SS), -2 at top of z_1 at end (George Garrett)
Typo: Solution SSLE.C52 had an extra 8 (David Braithwaite)
Typo: Exercise TSS.T40, added "consistent" in hypothesis (David Braithwaite)
Typo: Theorem MMCC, first justification in proof, reference corrected (Thomas Junier)
Typo: SAGE output delimiters confusing XML translation (Harald Schilly)
Typo: Exercise.SS.C44, Exercise.SS.C45, vector names fixed (George Garrett)
Typo: Example RSSC4 renamed properly to Example RSC4 (Section LDS) (George Garrett)
Typo: Variety of grammatical, spelling errors (George Garrett)
Typo: Variety of grammatical, spelling errors (Arnim Littek)
v2.00 2008/07/16
~~~~~~~~~~~~~
New: Waldron Edition as print-on-demand
New: jsMath electronic version
New: Exercise MINM.M80
New: Solution SD.T16 (Steve Canfield)
New: Notation AME and Notation GME (eigenvalue multiplicities)
Change: Theorem IMR upgraded to iff (Sam Antoine)
Change: Theorem IMILT, simplified proof using Theorem IMR
Change: Figures are now Diagrams - avoids clash with TeX figure macros
Change: Preface is no longer an Invariant Text, to conform with strict open standards
Typo: Definition LSMR, Notation LSMR renamed correctly as MRLS
Typo: Theorem NOILT, removed duplicate "injective" (Steve Canfield)
Typo: Replace "gratis" by "gratuit" in Preface (Hubert Dupont)
Typo: Section NLT, first paragraph, geometric/algebraic multiplicty (Mariana Montiel)
Edit: Minor edits throughout Reading Questions, based on Spring 2008 course
Edit: Updated Preface
v1.35 2008/04/16
~~~~~~~~~~~~~
New: Computation Notes for SAGE, initiated by Steve Canfield
New: Exercises B.T51, PD.T33, MR.T20
New: Exercise B.T50 (which may become a Theorem)
Typo: Solution EE.C20, replaced 2 by -2 in eigenvalue 3 computation (Taylor Firman)
Typo: Solution EE.C22, replaced four 3i's by \sqrt{3}i (Matt Salter)
Bug: Source for v1.34 wouldn't TeX easily. Recalled. (Chris Black)
v1.34 2008/03/17
~~~~~~~~~~~~~
New: Expanded Theorem NPNT to include converse
New: Table at end of Subsection CRS.CSSE summarizing related theorems
New: Figure CSRST
Typo: Removed stray "=" from statement of Theorem EMP (Greg Palmer)
Typo: Removed "rank" and "nullity", Annotated Acronyms, Chapter M (Steve Canfield)
Edit: Minor edits in Chapter M, based on Spring 2008 course
v1.33 2008/03/06
~~~~~~~~~~~~~
Maintenance release for figures
XML now contains figures
Figures included in source tarball, XML tarball
"List of Figures" still missing
Edit: Final revision of Fibonacci sequence, now Example FSCF (Section SD)
v1.32 2008/02/27
~~~~~~~~~~~~~
Figures now open-source via Python programs using PyX routines
This infrastructure may change, experimental right now
XML graphics import is not working
Needs a "List of Figures"
Not yet into source package, only in PDFs
New: Figures DTSLS, DLTA, DLTM, FTMR, FTMRA, MRCLT, GLT, ILT, NILT
Fixed excessive spacing following Proof Technique references
New: Use of Annotated Acronyms to Preface
New: Practicality and availability of hyperlinks to Preface
New: Exercises RREF.C33, NM.C50, NM.T30, LI.C32, LI.C33, LDS.C52, LISS.T50, O.T20, O.T30
New: Subsection SD.FS on Fibonacci sequences, needs some revision
Typo: Achetype C, second solution, x_2=7 (Betsy Kersey)
Typo: Example NSDS, portions of z_1 and z_3 reversed (Christina Roy)
Typo: Theorem TTMI, removed cruft from first half of proof (Christina Roy)
Typo: Example CSIP, carry -3+i through correctly (Matt Salter)
Edit: Example NRREF, named matrix E (from D) to avoid confusion with D, F as sets
Edit: Minor edits in Chapter R, concludes Fall 2007 course edits
Edit: Minor edits in Chapter SLE, Chapter V, based on Spring 2008 course
Edit: Streamlined proof of Theorem VRI
Edit: Definition of an Archetype is now from an open-source lexical database (WordNet)
v1.31 2008/02/02
~~~~~~~~~~~~~
Typo: Example VSP, b_1 corrected to a_1 (almost at the end)
Change: Rewrote proof of Theorem RREFU
Removed Subsection URREF from Section LC
Moved Theorem RREFU to Section RREF
Moved Techniques CD and U to precede
Change: Alternate proof of Theorem NMUS is now Exercise/Solution NM.T90
Edit: Clarified that Exercises C10-C19 in Section RREF are not designed to be worked by hand
Edit: Minor edits in Chapters VS, D, E, LT
Edit: Multiple edits in Section TSS
Edit: Removed unnecessary "m variables" from theorem hypotheses in Section TSS
Edit: Streamlined second half of proof for Theorem RCLS
Edit: Streamlined proof for Theorem SMZD
Edit: Streamlined proof for Theorem DRMM
New: Exercise SSLE.C51, SSLE.C52, RREF.T40
v1.30 2008/01/21
~~~~~~~~~~~~~
Production version for Spring 2008 UPS course
New: Annotated Acronyms: infrastructure is workable, but needs tuning
New: Annotated Acronyms for each of the 8 chapters in the core (Justin Platts)
New: Example CIVLT in Section IVLT, computing an inverse of a linear transformation
New: Exercise IVLT.M50
Change: Resolved acronym conflict for fields (commutativity, closure)
Property ACCN renamed to CACN, Property MCCN renamed to CMCN
Change: Solution vector is now Definition SOLV, conflicted with Singular Values
Edit: Technique L, clarified comments on object-oriented programming (Clive Spriggs)
Edit: Theorem ISRN, removed "m equations" from hypothesis
Edit: Theorem EMMVP, made proof more reliant on standard unit vectors
Edit: Reworked statement of Theorem LTDB to be clearer, edited later applications
Edit: Significantly shortened proof of Theorem VRLT with new version of Theorem LTDB
Edit: Minor edits in Chapters SLE, V, M
Edit: Minor edits to Reading Questions
Edit: Fixed a few dangling cross-references
New: Standard Unit Vectors (Definition SUV) formalized as notation, e_i
Obsolete: Theorem CBOB (Justin Platts)
As stated Theorem CBOB was likely false, proof was flawed
Correct version melded into UMCOB, which is now an equivalence
Some resulting adjustments in proof of Theorem PDM
v1.21 2007/11/21
~~~~~~~~~~~~~
New: Exercise RREF.C19, TSS.T41, PEE.T50
Maintenance release to fix XML on-line version
v1.20 2007/08/29
~~~~~~~~~~~~~
Production version for Fall 2007 UPS course
New: Exercises LI.T12, LI.T13, LT.T20, IVLT.T30
Typo: Solution MR.C42, employed inverse of representation (Will Swannack)
Concluded corrections from Spring 07 intro course (through CB)
v1.10
~~~~~~~~~~~~~
This version number skipped purposely
v1.09 2007/08/11
~~~~~~~~~~~~~
Obsolete: Exercise LC.T30 (duplicates Theorem PSHS)
New: Section HP, Hadamard product (by Elizabeth Million)
New: Exercises MO.T14, LISS.T40, PD.T40
Typo: Theorem PSMSR, forward implication, missing parentheses (Alex Paauwe)
Typo: Example SS6W, made S really 10 digits long by adding a leading zero (Elizabeth Million)
Typo: Section SAS, 4th paragraph, "choose m-1 coefficients" (Elizabeth Million)
Typo: Theorem DRCMA, second line of proof, remove row t of B (Tim Zitzer)
Typo: Solution LT.C30, switched roles of X_2, x_3 (Celia Ozereko)
Typo: Theorem ENLT, added "nilpotent" to hypothesis (Andrew Tapay)
Typo: Theorem PSHS, now zero vector, not zero scalar
Typo: Example COV, matrices I, A, B confused (Dylan Poulsen)
Typo: Theorem DEC, inserted sign changes (-1)^{i+j}
Change: Moved Definition SUV from Section MISLE to Section O
Change: Theorem SVD, fixed end of proof, reordered U and D (Tim Zitzer)
Change: Theorem DLDS, proof reworked in a more consistent style
Change: Theorem OSLI, proof reworked in a more consistent style
Change: Theorem EMMVP, proof reworked in a more consistent style
Change: Rewrote Solution PEE.T20
Change: Solution LISS.T40 now appropriate to location
Change: Example CROB4, basis justification changed (Will Swannack)
v1.08 2007/03/29
~~~~~~~~~~~~~
New: Section VM, Vandermonde Matrix
New: Section SAS, Sharing a Secret
New: Section CF, Curve Fitting (in-progress)
Change: Moved Section PSM out of Matrix Decompositions chapter
Typo: Corrected typo in Solution T.T30 (Andy Zimmer)
Typo: LISS.SS, color mixing analogy, changed "yellow" to "green" (Tate Urio)
Typo: Theorem COB, set T relabeled correctly to B (proof improved also)
v1.07 2007/03/14
~~~~~~~~~~~~~
New: Section SR, Square Root
New: Section POD, Polar Decomposition
New: Section SVD, Singular Value Decomposition
New: Added discussion of unitary matrices as isometries (Section OD)
New: Exercise PEE.T22
Change: Renamed Theorem GSP as Theorem GSPCV (Gram-Schmidt Procedure)
Change: Relabeled unitary matrices (from Q to U) in Section MINM
Section T, Trace (nee Section TM) in draft form
Chapter MD, Matrix Decompositions, outline is complete, all sections in draft form
v1.06 2007/03/08
~~~~~~~~~~~~~
New: Section PSM, Positive Semi-Definite Matrices
New: Section TM, Trace of a Matrix (by Andy Zimmer)
New: Theorem UMCOB in Section B, Theorem CBOB in Section CB
(Unitary matrices and orthonormal bases)
Typo: Example JCF10, removed extraneous identity linear transformations in matrix representations
Typo: Definition JCF, corrected last two portions (Kali Joseph)
Obsolete: Theorem VAC, Theorem CSSM, Theorem CVSM of Section VS never referenced
Removed and converted to Exercise VS.T21, VS.T22, VS.T23
Correction: Removed duplicate version of Theorem UTMR (Section JCF)
v1.05 2007/02/27
~~~~~~~~~~~~~
Change: Definition A (adjoint) moved into Section MO in new Subsection A
Change: Definition HM (Hermitian) moved into Section MM
New: Theorem CCM, Theorem AMA, Theorem AMSM, Theorem AA in Section MO
Basic results about adjoints and Hermitian matrices
New: Theorem MMAD, Theorem AIP, Theorem MMIP in Section MM
Basic results relating matrix multiplication, adjoints, inner products
New: Theorem OBNM (Section OD), orthonormal basis of eigenvectors for a normal matrix
Upgrade: Streamlined proof of Theorem HMRE (Section PEE)
Upgrade: Streamlined proof of Theorem HMOE (Section PEE)
Obsolete: \adj macro expunged in favor of \adjoint
Typo: Reading question #3 in MINM, "orthogonal" corrected to "unitary"
Typo: Proof of Theorem RREFU, last paragraph,
two subscripts corrected to read "except that b_k has a one in location d_k" (Bryan Smith)
v1.04 2007/02/15
~~~~~~~~~~~~~
New: Section OD, Orthonormal Diagonalization
Logical structure is complete, needs examples and editing
New: Example LTRGE in Section IS
New: More detailed proofs of Theorem CVRA and Theorem CRSM (Section O)
Typo: Theorem IPVA, added comma to vector list in statement (Celia Ozereko)
v1.03 2007/02/01
~~~~~~~~~~~~~
New: Section TD in draft form
New: Exercises RREF.C17,C18,C32; TSS.M56,M57,T40; HSE.C31; NM.M30
Typo: SSLE.SOL.C50 changed c>2 to c>1 (Celia Ozereko)
Addition: Reworded Definitions CVE, CVA, CVSM in Section VO to include C^m explicitly (Kyle Morril)
Change: Adjoint of a matrix notation is now a superscript star, changes only one occurence
Change: Subsection SD.OD (Orthonormal Diagonalization) downgraded to a comment
Edit: Minor editing in Section MINM (mostly source-level changes)
v1.02 2007/01/25
~~~~~~~~~~~~~
New: Section JCF essentially complete, now a draft
New: Section ROD in draft form
New: Solution to MR.T80
New: Example JCF10 in Section JCF
Typo: Reference in Exercise MR.T80 fixed to read "Theorem EMP"
Typo: Commutative diagram after Example MR.MPMR
basis on final matrix representation now D (not C) (Tim Zitzer)
Typo: HSE.SOL.T10, implications reversed (Tim Zitzer)