a first course in linear algebra - an open-source textbook
TRANSCRIPT
![Page 1: A First Course in Linear Algebra - An Open-Source Textbook](https://reader036.vdocuments.us/reader036/viewer/2022070222/613d2171736caf36b759ad49/html5/thumbnails/1.jpg)
A First Course in Linear AlgebraAn Open-Source Textbook
Department of Mathematics and Computer ScienceUniversity of Puget Sound
Sage Developer Days 1University of Washington
June 16, 2008
![Page 2: A First Course in Linear Algebra - An Open-Source Textbook](https://reader036.vdocuments.us/reader036/viewer/2022070222/613d2171736caf36b759ad49/html5/thumbnails/2.jpg)
Introduction
Overview
A free (no cost) introductory textbook
A free (GFDL’ed) introductory textbook
Designed to encourage modification
Designed to encourage content contributions
A social experiment
A disruption to traditional publishing
Today’s Talk
Open source
Linear algebra
SAGE enhanced version
Parallels to SAGE development
A wee bit of Python
Rob Beezer (U Puget Sound) A First Course in Linear Algebra Sage Dev1 June 16, 2008 2 / 26
![Page 3: A First Course in Linear Algebra - An Open-Source Textbook](https://reader036.vdocuments.us/reader036/viewer/2022070222/613d2171736caf36b759ad49/html5/thumbnails/3.jpg)
Introduction
Overview
A free (no cost) introductory textbook
A free (GFDL’ed) introductory textbook
Designed to encourage modification
Designed to encourage content contributions
A social experiment
A disruption to traditional publishing
Today’s Talk
Open source
Linear algebra
SAGE enhanced version
Parallels to SAGE development
A wee bit of Python
Rob Beezer (U Puget Sound) A First Course in Linear Algebra Sage Dev1 June 16, 2008 2 / 26
![Page 4: A First Course in Linear Algebra - An Open-Source Textbook](https://reader036.vdocuments.us/reader036/viewer/2022070222/613d2171736caf36b759ad49/html5/thumbnails/4.jpg)
Introduction
Table of ContentsCore
I Systems of Linear EquationsI VectorsI MatricesI Vector SpacesI DeterminantsI EigenvaluesI Linear TransformationsI Representations
TopicsI Positive Semi-Definite MatricesI Singular Value DecompositionI And more . . .
ApplicationsI Curve FittingI Secret SharingI 〈Your Contribution Here〉
Rob Beezer (U Puget Sound) A First Course in Linear Algebra Sage Dev1 June 16, 2008 3 / 26
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Introduction
By The Numbers
7 Chapters (Core)
43 Sections (Core)
10 Topics
2 Applications
133 Definitions
257 Theorems
382 Exercises
254 Solutions
24 Archetypes
Rob Beezer (U Puget Sound) A First Course in Linear Algebra Sage Dev1 June 16, 2008 4 / 26
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Motivation
Why?
Classic yellowed notes with many additions in the margins
Moved to TEX version for convenience (and legibility)
Began passing out to students, quality improved
Abstract Algebra textbook went to a new edition
Linear Algebra textbook in a new edition soon
Combinatorics textbook out-of-print
Unnecessary revisions drive up costs
Would open-source software model carry over?
Scratching an itch
“The world does not need another linear algebra text,the world does needs a free linear algebra text.”
Lead by example
Rob Beezer (U Puget Sound) A First Course in Linear Algebra Sage Dev1 June 16, 2008 5 / 26
![Page 7: A First Course in Linear Algebra - An Open-Source Textbook](https://reader036.vdocuments.us/reader036/viewer/2022070222/613d2171736caf36b759ad49/html5/thumbnails/7.jpg)
Motivation
Why?
Classic yellowed notes with many additions in the margins
Moved to TEX version for convenience (and legibility)
Began passing out to students, quality improved
Abstract Algebra textbook went to a new edition
Linear Algebra textbook in a new edition soon
Combinatorics textbook out-of-print
Unnecessary revisions drive up costs
Would open-source software model carry over?
Scratching an itch
“The world does not need another linear algebra text,the world does needs a free linear algebra text.”
Lead by example
Rob Beezer (U Puget Sound) A First Course in Linear Algebra Sage Dev1 June 16, 2008 5 / 26
![Page 8: A First Course in Linear Algebra - An Open-Source Textbook](https://reader036.vdocuments.us/reader036/viewer/2022070222/613d2171736caf36b759ad49/html5/thumbnails/8.jpg)
Motivation
Why?
Classic yellowed notes with many additions in the margins
Moved to TEX version for convenience (and legibility)
Began passing out to students, quality improved
Abstract Algebra textbook went to a new edition
Linear Algebra textbook in a new edition soon
Combinatorics textbook out-of-print
Unnecessary revisions drive up costs
Would open-source software model carry over?
Scratching an itch
“The world does not need another linear algebra text,the world does needs a free linear algebra text.”
Lead by example
Rob Beezer (U Puget Sound) A First Course in Linear Algebra Sage Dev1 June 16, 2008 5 / 26
![Page 9: A First Course in Linear Algebra - An Open-Source Textbook](https://reader036.vdocuments.us/reader036/viewer/2022070222/613d2171736caf36b759ad49/html5/thumbnails/9.jpg)
Motivation
Short History
September 2002: moved from paper to electronic notes
January 2004: in earnest, with a complete restart
December 2004: rough draft complete
December 2006: Version 1.0
January 2008: Used at Miramar College, St. Cloud State U
Summer 2008: Version 2.0
Rob Beezer (U Puget Sound) A First Course in Linear Algebra Sage Dev1 June 16, 2008 6 / 26
![Page 10: A First Course in Linear Algebra - An Open-Source Textbook](https://reader036.vdocuments.us/reader036/viewer/2022070222/613d2171736caf36b759ad49/html5/thumbnails/10.jpg)
Motivation
Short History
September 2002: moved from paper to electronic notes
January 2004: in earnest, with a complete restart
December 2004: rough draft complete
December 2006: Version 1.0
January 2008: Used at Miramar College, St. Cloud State U
Summer 2008: Version 2.0
Rob Beezer (U Puget Sound) A First Course in Linear Algebra Sage Dev1 June 16, 2008 6 / 26
![Page 11: A First Course in Linear Algebra - An Open-Source Textbook](https://reader036.vdocuments.us/reader036/viewer/2022070222/613d2171736caf36b759ad49/html5/thumbnails/11.jpg)
A Closer Look
Targets
Not an online course, or online textbookI Internet for distribution: nearly zero costI Internet for marketing: Google “linear algebra,” Page 1I Web site: Approx 6,000 monthly visitors
Production script in Python creates . . .
Downloadable PDF in a variety of formatsI US Letter, A4I One-sided, two-sidedI Screen format (4:3, 16:9)
Print-on-demand, Lulu.com, $24.95
XML/MathML
jsMath (next week?)
SAGE worksheets (planned)
Theorem and Definition flashcards
Tarballs for LATEX source, XML (jsMath, SAGE worksheets)
Rob Beezer (U Puget Sound) A First Course in Linear Algebra Sage Dev1 June 16, 2008 7 / 26
![Page 12: A First Course in Linear Algebra - An Open-Source Textbook](https://reader036.vdocuments.us/reader036/viewer/2022070222/613d2171736caf36b759ad49/html5/thumbnails/12.jpg)
A Closer Look
Targets
Not an online course, or online textbookI Internet for distribution: nearly zero costI Internet for marketing: Google “linear algebra,” Page 1I Web site: Approx 6,000 monthly visitors
Production script in Python creates . . .
Downloadable PDF in a variety of formatsI US Letter, A4I One-sided, two-sidedI Screen format (4:3, 16:9)
Print-on-demand, Lulu.com, $24.95
XML/MathML
jsMath (next week?)
SAGE worksheets (planned)
Theorem and Definition flashcards
Tarballs for LATEX source, XML (jsMath, SAGE worksheets)
Rob Beezer (U Puget Sound) A First Course in Linear Algebra Sage Dev1 June 16, 2008 7 / 26
![Page 13: A First Course in Linear Algebra - An Open-Source Textbook](https://reader036.vdocuments.us/reader036/viewer/2022070222/613d2171736caf36b759ad49/html5/thumbnails/13.jpg)
A Closer Look
Targets
Not an online course, or online textbookI Internet for distribution: nearly zero costI Internet for marketing: Google “linear algebra,” Page 1I Web site: Approx 6,000 monthly visitors
Production script in Python creates . . .
Downloadable PDF in a variety of formatsI US Letter, A4I One-sided, two-sidedI Screen format (4:3, 16:9)
Print-on-demand, Lulu.com, $24.95
XML/MathML
jsMath (next week?)
SAGE worksheets (planned)
Theorem and Definition flashcards
Tarballs for LATEX source, XML (jsMath, SAGE worksheets)
Rob Beezer (U Puget Sound) A First Course in Linear Algebra Sage Dev1 June 16, 2008 7 / 26
![Page 14: A First Course in Linear Algebra - An Open-Source Textbook](https://reader036.vdocuments.us/reader036/viewer/2022070222/613d2171736caf36b759ad49/html5/thumbnails/14.jpg)
A Closer Look
Targets
Not an online course, or online textbookI Internet for distribution: nearly zero costI Internet for marketing: Google “linear algebra,” Page 1I Web site: Approx 6,000 monthly visitors
Production script in Python creates . . .
Downloadable PDF in a variety of formatsI US Letter, A4I One-sided, two-sidedI Screen format (4:3, 16:9)
Print-on-demand, Lulu.com, $24.95
XML/MathML
jsMath (next week?)
SAGE worksheets (planned)
Theorem and Definition flashcards
Tarballs for LATEX source, XML (jsMath, SAGE worksheets)
Rob Beezer (U Puget Sound) A First Course in Linear Algebra Sage Dev1 June 16, 2008 7 / 26
![Page 15: A First Course in Linear Algebra - An Open-Source Textbook](https://reader036.vdocuments.us/reader036/viewer/2022070222/613d2171736caf36b759ad49/html5/thumbnails/15.jpg)
A Closer Look
Targets
Not an online course, or online textbookI Internet for distribution: nearly zero costI Internet for marketing: Google “linear algebra,” Page 1I Web site: Approx 6,000 monthly visitors
Production script in Python creates . . .
Downloadable PDF in a variety of formatsI US Letter, A4I One-sided, two-sidedI Screen format (4:3, 16:9)
Print-on-demand, Lulu.com, $24.95
XML/MathML
jsMath (next week?)
SAGE worksheets (planned)
Theorem and Definition flashcards
Tarballs for LATEX source, XML (jsMath, SAGE worksheets)
Rob Beezer (U Puget Sound) A First Course in Linear Algebra Sage Dev1 June 16, 2008 7 / 26
![Page 16: A First Course in Linear Algebra - An Open-Source Textbook](https://reader036.vdocuments.us/reader036/viewer/2022070222/613d2171736caf36b759ad49/html5/thumbnails/16.jpg)
A Closer Look
Targets
Not an online course, or online textbookI Internet for distribution: nearly zero costI Internet for marketing: Google “linear algebra,” Page 1I Web site: Approx 6,000 monthly visitors
Production script in Python creates . . .
Downloadable PDF in a variety of formatsI US Letter, A4I One-sided, two-sidedI Screen format (4:3, 16:9)
Print-on-demand, Lulu.com, $24.95
XML/MathML
jsMath (next week?)
SAGE worksheets (planned)
Theorem and Definition flashcards
Tarballs for LATEX source, XML (jsMath, SAGE worksheets)
Rob Beezer (U Puget Sound) A First Course in Linear Algebra Sage Dev1 June 16, 2008 7 / 26
![Page 17: A First Course in Linear Algebra - An Open-Source Textbook](https://reader036.vdocuments.us/reader036/viewer/2022070222/613d2171736caf36b759ad49/html5/thumbnails/17.jpg)
A Closer Look
Targets
Not an online course, or online textbookI Internet for distribution: nearly zero costI Internet for marketing: Google “linear algebra,” Page 1I Web site: Approx 6,000 monthly visitors
Production script in Python creates . . .
Downloadable PDF in a variety of formatsI US Letter, A4I One-sided, two-sidedI Screen format (4:3, 16:9)
Print-on-demand, Lulu.com, $24.95
XML/MathML
jsMath (next week?)
SAGE worksheets (planned)
Theorem and Definition flashcards
Tarballs for LATEX source, XML (jsMath, SAGE worksheets)
Rob Beezer (U Puget Sound) A First Course in Linear Algebra Sage Dev1 June 16, 2008 7 / 26
![Page 18: A First Course in Linear Algebra - An Open-Source Textbook](https://reader036.vdocuments.us/reader036/viewer/2022070222/613d2171736caf36b759ad49/html5/thumbnails/18.jpg)
A Closer Look
Targets
Not an online course, or online textbookI Internet for distribution: nearly zero costI Internet for marketing: Google “linear algebra,” Page 1I Web site: Approx 6,000 monthly visitors
Production script in Python creates . . .
Downloadable PDF in a variety of formatsI US Letter, A4I One-sided, two-sidedI Screen format (4:3, 16:9)
Print-on-demand, Lulu.com, $24.95
XML/MathML
jsMath (next week?)
SAGE worksheets (planned)
Theorem and Definition flashcards
Tarballs for LATEX source, XML (jsMath, SAGE worksheets)
Rob Beezer (U Puget Sound) A First Course in Linear Algebra Sage Dev1 June 16, 2008 7 / 26
![Page 19: A First Course in Linear Algebra - An Open-Source Textbook](https://reader036.vdocuments.us/reader036/viewer/2022070222/613d2171736caf36b759ad49/html5/thumbnails/19.jpg)
A Closer Look
Under the Hood
Primary Tool: LATEX + pdflatex
Packages: AMS, hyperref, tocloft
Translator: tex4ht to build MathML and jsMath
Graphics: PyX + Python to build PDF
Sectioning: Custom macros override/extend LATEX
References: Custom macros override/extend LATEX, hyperref
Rob Beezer (U Puget Sound) A First Course in Linear Algebra Sage Dev1 June 16, 2008 8 / 26
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A Closer Look
Acronyms
Every definition, theorem, chapter, section, figure, exercise is labeled witha mnemonic acronym (five letter maximum).
Promotes customization (e.g. deleting sections)
“Theorem SMZD” preferable to “Theorem 8.3.2”
Macro: \acronymref{theorem}{SMZD} for easy referencing
Unexpected benefits for text processing
Rob Beezer (U Puget Sound) A First Course in Linear Algebra Sage Dev1 June 16, 2008 9 / 26
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A Closer Look
Typical PDF
Rob Beezer (U Puget Sound) A First Course in Linear Algebra Sage Dev1 June 16, 2008 10 / 26
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A Closer Look
Typical Front Matter
Rob Beezer (U Puget Sound) A First Course in Linear Algebra Sage Dev1 June 16, 2008 11 / 26
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A Closer Look
Typical XML/MathML
Rob Beezer (U Puget Sound) A First Course in Linear Algebra Sage Dev1 June 16, 2008 12 / 26
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A Closer Look
Typical jsMath
Rob Beezer (U Puget Sound) A First Course in Linear Algebra Sage Dev1 June 16, 2008 13 / 26
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A Closer Look
Prototypical SAGE Worksheet
Rob Beezer (U Puget Sound) A First Course in Linear Algebra Sage Dev1 June 16, 2008 14 / 26
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A Closer Look
Typical Figure
u
ρB (u)
T (u) = ρ−1C
(MT
B,CρB (u))
MTB,CρB (u)
T
MTB,C
ρB ρ−1C
Rob Beezer (U Puget Sound) A First Course in Linear Algebra Sage Dev1 June 16, 2008 15 / 26
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A Closer Look
Figure Source, Python + PyX
Rob Beezer (U Puget Sound) A First Course in Linear Algebra Sage Dev1 June 16, 2008 16 / 26
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Fun with Text
Flash Cards
Simple sed script extracts every definition and theorem
Reformat as two 4′′ × 6′′ cards per page
Rob Beezer (U Puget Sound) A First Course in Linear Algebra Sage Dev1 June 16, 2008 17 / 26
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Fun with Text
Theorem and Definition Dependencies
Python script to parse LATEX source
Record each definition and theorem acronym as a node
For each theorem, note acronyms used in proof
Build directed edges from theorems to their prerequisites
Draw graph with GraphViz
Analyze resultant graph: high degree?, root?, longest path?
Rob Beezer (U Puget Sound) A First Course in Linear Algebra Sage Dev1 June 16, 2008 18 / 26
![Page 30: A First Course in Linear Algebra - An Open-Source Textbook](https://reader036.vdocuments.us/reader036/viewer/2022070222/613d2171736caf36b759ad49/html5/thumbnails/30.jpg)
Fun with Text
Theorem and Definition Dependencies
Python script to parse LATEX source
Record each definition and theorem acronym as a node
For each theorem, note acronyms used in proof
Build directed edges from theorems to their prerequisites
Draw graph with GraphViz
Analyze resultant graph: high degree?, root?, longest path?
Rob Beezer (U Puget Sound) A First Course in Linear Algebra Sage Dev1 June 16, 2008 18 / 26
![Page 31: A First Course in Linear Algebra - An Open-Source Textbook](https://reader036.vdocuments.us/reader036/viewer/2022070222/613d2171736caf36b759ad49/html5/thumbnails/31.jpg)
Fun with Text
Theorem and Definition Dependencies
Python script to parse LATEX source
Record each definition and theorem acronym as a node
For each theorem, note acronyms used in proof
Build directed edges from theorems to their prerequisites
Draw graph with GraphViz
Analyze resultant graph: high degree?, root?, longest path?
Rob Beezer (U Puget Sound) A First Course in Linear Algebra Sage Dev1 June 16, 2008 18 / 26
![Page 32: A First Course in Linear Algebra - An Open-Source Textbook](https://reader036.vdocuments.us/reader036/viewer/2022070222/613d2171736caf36b759ad49/html5/thumbnails/32.jpg)
Open Source Design
Customizable
Designed to be customized
Extensive use of macrosI Notation: \newcommand[1]{\transpose}{#1ˆ t}I \transpose{(AB)}=\transpose{B}\transpose{A}
Even diagrams/figures have editable source code
Switches control format, organization, inclusion (keyval package)
Granular, 1250 files
Benefits for adapting to translators
Benefits for text processing
Rob Beezer (U Puget Sound) A First Course in Linear Algebra Sage Dev1 June 16, 2008 19 / 26
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Open Source Design
Extensible
Exercises and SolutionsI Drop-in (almost)I Suggested by Debian configurations
Topics and ApplicationsI Trace, by Andy ZimmerI Hadamard Product, by Elizabeth Million
Computation NotesI TI-86 calculatorI MathematicaI SAGE
Still, as Benevolent Dictator, strict control on Core
Rob Beezer (U Puget Sound) A First Course in Linear Algebra Sage Dev1 June 16, 2008 20 / 26
![Page 34: A First Course in Linear Algebra - An Open-Source Textbook](https://reader036.vdocuments.us/reader036/viewer/2022070222/613d2171736caf36b759ad49/html5/thumbnails/34.jpg)
Open Source Design
Extensible
Exercises and SolutionsI Drop-in (almost)I Suggested by Debian configurations
Topics and ApplicationsI Trace, by Andy ZimmerI Hadamard Product, by Elizabeth Million
Computation NotesI TI-86 calculatorI MathematicaI SAGE
Still, as Benevolent Dictator, strict control on Core
Rob Beezer (U Puget Sound) A First Course in Linear Algebra Sage Dev1 June 16, 2008 20 / 26
![Page 35: A First Course in Linear Algebra - An Open-Source Textbook](https://reader036.vdocuments.us/reader036/viewer/2022070222/613d2171736caf36b759ad49/html5/thumbnails/35.jpg)
Open Source Design
Extensible
Exercises and SolutionsI Drop-in (almost)I Suggested by Debian configurations
Topics and ApplicationsI Trace, by Andy ZimmerI Hadamard Product, by Elizabeth Million
Computation NotesI TI-86 calculatorI MathematicaI SAGE
Still, as Benevolent Dictator, strict control on Core
Rob Beezer (U Puget Sound) A First Course in Linear Algebra Sage Dev1 June 16, 2008 20 / 26
![Page 36: A First Course in Linear Algebra - An Open-Source Textbook](https://reader036.vdocuments.us/reader036/viewer/2022070222/613d2171736caf36b759ad49/html5/thumbnails/36.jpg)
Open Source Design
Extensible
Exercises and SolutionsI Drop-in (almost)I Suggested by Debian configurations
Topics and ApplicationsI Trace, by Andy ZimmerI Hadamard Product, by Elizabeth Million
Computation NotesI TI-86 calculatorI MathematicaI SAGE
Still, as Benevolent Dictator, strict control on Core
Rob Beezer (U Puget Sound) A First Course in Linear Algebra Sage Dev1 June 16, 2008 20 / 26
![Page 37: A First Course in Linear Algebra - An Open-Source Textbook](https://reader036.vdocuments.us/reader036/viewer/2022070222/613d2171736caf36b759ad49/html5/thumbnails/37.jpg)
Mathematics, Computing, Teaching
Some Linear Algebra
Problem: Column space of A =
[1 2 32 4 6
]?
Answer: All multiples of the first column, a dimension 1 subspace
A common textbook approach:
Column space is vectors b such that Ax = b is consistent
So row-reduce augmented matrix [A|b] to study solutions[1 2 3 b1
2 4 6 b2
]→[
1 2 3 b1
0 0 0 −2b1 + b2
]b in column space ⇐⇒ system consistent ⇐⇒ −2b1 + b2 = 0
Column space vectors are solutions to a homogeneous system
Rob Beezer (U Puget Sound) A First Course in Linear Algebra Sage Dev1 June 16, 2008 21 / 26
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Mathematics, Computing, Teaching
Some Linear Algebra
Problem: Column space of A =
[1 2 32 4 6
]?
Answer: All multiples of the first column, a dimension 1 subspace
A common textbook approach:
Column space is vectors b such that Ax = b is consistent
So row-reduce augmented matrix [A|b] to study solutions[1 2 3 b1
2 4 6 b2
]→[
1 2 3 b1
0 0 0 −2b1 + b2
]b in column space ⇐⇒ system consistent ⇐⇒ −2b1 + b2 = 0
Column space vectors are solutions to a homogeneous system
Rob Beezer (U Puget Sound) A First Course in Linear Algebra Sage Dev1 June 16, 2008 21 / 26
![Page 39: A First Course in Linear Algebra - An Open-Source Textbook](https://reader036.vdocuments.us/reader036/viewer/2022070222/613d2171736caf36b759ad49/html5/thumbnails/39.jpg)
Mathematics, Computing, Teaching
CAS Assistance
Row-reduce:
[1 2 3 b1
2 4 6 b2
]→[
1 2 3 00 0 0 1
]
Work-around:
[1 2 3 b1 02 4 6 0 b2
]→
[1 2 3 0 b2
2
0 0 0 1 − b22b1
]Scale row 2: →
[1 2 3 0 b2
20 0 0 −2b1 b2
]Add last two columns:
Second row: −2b1 + b2; equals zero ⇐⇒ b in column space
First row: b22 = b1 when b is in column space
Better:
[1 2 3 1 02 4 6 0 1
]→[
1 2 3 0 12
0 0 0 1 −12
]Column space of A is null space of
[1 −1
2
]
Rob Beezer (U Puget Sound) A First Course in Linear Algebra Sage Dev1 June 16, 2008 22 / 26
![Page 40: A First Course in Linear Algebra - An Open-Source Textbook](https://reader036.vdocuments.us/reader036/viewer/2022070222/613d2171736caf36b759ad49/html5/thumbnails/40.jpg)
Mathematics, Computing, Teaching
CAS Assistance
Row-reduce:
[1 2 3 b1
2 4 6 b2
]→[
1 2 3 00 0 0 1
]
Work-around:
[1 2 3 b1 02 4 6 0 b2
]→
[1 2 3 0 b2
2
0 0 0 1 − b22b1
]Scale row 2: →
[1 2 3 0 b2
20 0 0 −2b1 b2
]Add last two columns:
Second row: −2b1 + b2; equals zero ⇐⇒ b in column space
First row: b22 = b1 when b is in column space
Better:
[1 2 3 1 02 4 6 0 1
]→[
1 2 3 0 12
0 0 0 1 −12
]Column space of A is null space of
[1 −1
2
]
Rob Beezer (U Puget Sound) A First Course in Linear Algebra Sage Dev1 June 16, 2008 22 / 26
![Page 41: A First Course in Linear Algebra - An Open-Source Textbook](https://reader036.vdocuments.us/reader036/viewer/2022070222/613d2171736caf36b759ad49/html5/thumbnails/41.jpg)
Mathematics, Computing, Teaching
CAS Assistance
Row-reduce:
[1 2 3 b1
2 4 6 b2
]→[
1 2 3 00 0 0 1
]
Work-around:
[1 2 3 b1 02 4 6 0 b2
]→
[1 2 3 0 b2
2
0 0 0 1 − b22b1
]Scale row 2: →
[1 2 3 0 b2
20 0 0 −2b1 b2
]Add last two columns:
Second row: −2b1 + b2; equals zero ⇐⇒ b in column space
First row: b22 = b1 when b is in column space
Better:
[1 2 3 1 02 4 6 0 1
]→[
1 2 3 0 12
0 0 0 1 −12
]Column space of A is null space of
[1 −1
2
]Rob Beezer (U Puget Sound) A First Course in Linear Algebra Sage Dev1 June 16, 2008 22 / 26
![Page 42: A First Course in Linear Algebra - An Open-Source Textbook](https://reader036.vdocuments.us/reader036/viewer/2022070222/613d2171736caf36b759ad49/html5/thumbnails/42.jpg)
Mathematics, Computing, Teaching
Extended Echelon Form
A m × n matrix[A|Im] augment with m ×m identity matrix
Row-reduce to echelon form: [A|Im]→ [B|J] “extended echelon form”
Then
B is echelon form of A
J is nonsingular
J is product of elementary matrices for row operations
B = JA
Ax = y ⇐⇒ Bx = Jy
A nonsingular ⇒ J = A−1
Rob Beezer (U Puget Sound) A First Course in Linear Algebra Sage Dev1 June 16, 2008 23 / 26
![Page 43: A First Course in Linear Algebra - An Open-Source Textbook](https://reader036.vdocuments.us/reader036/viewer/2022070222/613d2171736caf36b759ad49/html5/thumbnails/43.jpg)
Mathematics, Computing, Teaching
Extended Echelon Form
A m × n matrix[A|Im] augment with m ×m identity matrix
Row-reduce to echelon form: [A|Im]→ [B|J] “extended echelon form”
Then
B is echelon form of A
J is nonsingular
J is product of elementary matrices for row operations
B = JA
Ax = y ⇐⇒ Bx = Jy
A nonsingular ⇒ J = A−1
Rob Beezer (U Puget Sound) A First Course in Linear Algebra Sage Dev1 June 16, 2008 23 / 26
![Page 44: A First Course in Linear Algebra - An Open-Source Textbook](https://reader036.vdocuments.us/reader036/viewer/2022070222/613d2171736caf36b759ad49/html5/thumbnails/44.jpg)
Mathematics, Computing, Teaching
Four Subspaces
[A|Im]→ [B|J] =
[C K
0 L
]Let r be the rank of A. L is an (m − r)×m matrix in echelon form and
Row space of A is row space of C
Null space of A is null space of C
Column space of A is null space of L
Left null space of A is row space of L
Proof:
B = JA is key tool
No dimension arguments
No vector space properties of subspaces
Corollary: Subspace dimensions in terms of n, m and r
Rob Beezer (U Puget Sound) A First Course in Linear Algebra Sage Dev1 June 16, 2008 24 / 26
![Page 45: A First Course in Linear Algebra - An Open-Source Textbook](https://reader036.vdocuments.us/reader036/viewer/2022070222/613d2171736caf36b759ad49/html5/thumbnails/45.jpg)
Mathematics, Computing, Teaching
Four Subspaces
[A|Im]→ [B|J] =
[C K
0 L
]Let r be the rank of A. L is an (m − r)×m matrix in echelon form and
Row space of A is row space of C
Null space of A is null space of C
Column space of A is null space of L
Left null space of A is row space of L
Proof:
B = JA is key tool
No dimension arguments
No vector space properties of subspaces
Corollary: Subspace dimensions in terms of n, m and r
Rob Beezer (U Puget Sound) A First Course in Linear Algebra Sage Dev1 June 16, 2008 24 / 26
![Page 46: A First Course in Linear Algebra - An Open-Source Textbook](https://reader036.vdocuments.us/reader036/viewer/2022070222/613d2171736caf36b759ad49/html5/thumbnails/46.jpg)
Mathematics, Computing, Teaching
Four Subspaces
[A|Im]→ [B|J] =
[C K
0 L
]Let r be the rank of A. L is an (m − r)×m matrix in echelon form and
Row space of A is row space of C
Null space of A is null space of C
Column space of A is null space of L
Left null space of A is row space of L
Proof:
B = JA is key tool
No dimension arguments
No vector space properties of subspaces
Corollary: Subspace dimensions in terms of n, m and r
Rob Beezer (U Puget Sound) A First Course in Linear Algebra Sage Dev1 June 16, 2008 24 / 26
![Page 47: A First Course in Linear Algebra - An Open-Source Textbook](https://reader036.vdocuments.us/reader036/viewer/2022070222/613d2171736caf36b759ad49/html5/thumbnails/47.jpg)
Mathematics, Computing, Teaching
Four Subspaces
[A|Im]→ [B|J] =
[C K
0 L
]Let r be the rank of A. L is an (m − r)×m matrix in echelon form and
Row space of A is row space of C
Null space of A is null space of C
Column space of A is null space of L
Left null space of A is row space of L
Proof:
B = JA is key tool
No dimension arguments
No vector space properties of subspaces
Corollary: Subspace dimensions in terms of n, m and r
Rob Beezer (U Puget Sound) A First Course in Linear Algebra Sage Dev1 June 16, 2008 24 / 26
![Page 48: A First Course in Linear Algebra - An Open-Source Textbook](https://reader036.vdocuments.us/reader036/viewer/2022070222/613d2171736caf36b759ad49/html5/thumbnails/48.jpg)
Conclusion
Open-Source Benefits
No deadlines!
Rapid correction of typos and frequent releases
Unparalled navigation in electronic editions (no DRM!)
Text processing of source for derivative supplements
Classroom-inspired exercises
Student contributions
tex4ht: LATEX→XML→Braille
Introduced to SAGE (contributed Computation Notes)
Interesting contributorsI Frenchman correcting my discourse on “liberte” versus “gratis”I Retired South African mining engineer coding echelon form() routineI Indonesian lecturer commenting on finer points of a proof
Rob Beezer (U Puget Sound) A First Course in Linear Algebra Sage Dev1 June 16, 2008 25 / 26
![Page 49: A First Course in Linear Algebra - An Open-Source Textbook](https://reader036.vdocuments.us/reader036/viewer/2022070222/613d2171736caf36b759ad49/html5/thumbnails/49.jpg)
Conclusion
Projects
Version control repository
Forum for student discussions
E-Book formats (Kindle, Sony), re-flowable?
Conversion to speech? (tex4ht)
Automate: jsMath pages to SAGE worksheetsI Linking across worksheetsI SAGE code: LATEX source→jsMath→Worksheet
Automate: creating “textbook” exercisesI SAGE codeI Web form interfaceI Request rows, columns, rank, specific pivot columnsI Output matrices with “nice” bases for all 4 subspacesI Or, request Jordan canonical form
Rob Beezer (U Puget Sound) A First Course in Linear Algebra Sage Dev1 June 16, 2008 26 / 26
![Page 50: A First Course in Linear Algebra - An Open-Source Textbook](https://reader036.vdocuments.us/reader036/viewer/2022070222/613d2171736caf36b759ad49/html5/thumbnails/50.jpg)
Conclusion
Projects
Version control repository
Forum for student discussions
E-Book formats (Kindle, Sony), re-flowable?
Conversion to speech? (tex4ht)
Automate: jsMath pages to SAGE worksheetsI Linking across worksheetsI SAGE code: LATEX source→jsMath→Worksheet
Automate: creating “textbook” exercisesI SAGE codeI Web form interfaceI Request rows, columns, rank, specific pivot columnsI Output matrices with “nice” bases for all 4 subspacesI Or, request Jordan canonical form
Rob Beezer (U Puget Sound) A First Course in Linear Algebra Sage Dev1 June 16, 2008 26 / 26
![Page 51: A First Course in Linear Algebra - An Open-Source Textbook](https://reader036.vdocuments.us/reader036/viewer/2022070222/613d2171736caf36b759ad49/html5/thumbnails/51.jpg)
Conclusion
Projects
Version control repository
Forum for student discussions
E-Book formats (Kindle, Sony), re-flowable?
Conversion to speech? (tex4ht)
Automate: jsMath pages to SAGE worksheetsI Linking across worksheetsI SAGE code: LATEX source→jsMath→Worksheet
Automate: creating “textbook” exercisesI SAGE codeI Web form interfaceI Request rows, columns, rank, specific pivot columnsI Output matrices with “nice” bases for all 4 subspacesI Or, request Jordan canonical form
Rob Beezer (U Puget Sound) A First Course in Linear Algebra Sage Dev1 June 16, 2008 26 / 26
![Page 52: A First Course in Linear Algebra - An Open-Source Textbook](https://reader036.vdocuments.us/reader036/viewer/2022070222/613d2171736caf36b759ad49/html5/thumbnails/52.jpg)
Conclusion
Projects
Version control repository
Forum for student discussions
E-Book formats (Kindle, Sony), re-flowable?
Conversion to speech? (tex4ht)
Automate: jsMath pages to SAGE worksheetsI Linking across worksheetsI SAGE code: LATEX source→jsMath→Worksheet
Automate: creating “textbook” exercisesI SAGE codeI Web form interfaceI Request rows, columns, rank, specific pivot columnsI Output matrices with “nice” bases for all 4 subspacesI Or, request Jordan canonical form
Rob Beezer (U Puget Sound) A First Course in Linear Algebra Sage Dev1 June 16, 2008 26 / 26