A Modern Online Linear Algebra Textbook
A Modern Online Linear Algebra TextbookRob Beezerbeezer@pugetsound.eduDepartment of Mathematics and Computer ScienceUniversity of Puget Sound18th Conferenceof theInternational Linear Algebra SocietyProvidence, Rhode IslandJune 7, 2013
Introduction and OutlineTwo parts: Thoughts on organizing an introductory course Modern approach to textbook design and distribution Follow along in the third half:http://linear.ups.edu, left sidebar: “Online” Support: NSF TUES Grant, UTMOST project, utmost.aimath.org Support: Shuttleworth Foundation Flash GrantRob Beezer (U Puget Sound)A First Course in Linear AlgebraILASJune 20132 / 22
A First Course in Linear Algebra Initiated 2003; Version 1.0 2006 Always free online GNU Free Documentation License Sophomore course Emphasis on proof techniquesRob Beezer (U Puget Sound)A First Course in Linear AlgebraILASJune 20133 / 22
Chapter: System of Equations Best motivation for students coming out of calculus Hint: reduced row-echelon form is a column-by-column algorithm Natural place to introduce null spaces and nonsingular matrices Cycle back and rephrase in the language of the linear transformationT : Cn CmRob Beezer (U Puget Sound)T (x) AxA First Course in Linear AlgebraILASJune 20134 / 22
Chapter: Vectors A vector space has addition and scalar multiplication So a linear combination is the most natural construction Spanning sets and linear independence followRob Beezer (U Puget Sound)A First Course in Linear AlgebraILASJune 20135 / 22
Chapter: Vectors Other consequences: Product of a matrix A and a vector x is the linear combination of thecolumns of A with scalars from the entries of x Matrix multiplication:AB A[B1 B2 . . . Bp ] [AB1 AB2 . . . ABp ] The entry-by-entry formula for a matrix product,Xaij bjkjis now a theorem, derived from linear combinationsRob Beezer (U Puget Sound)A First Course in Linear AlgebraILASJune 20136 / 22
Chapter: Matrices Matrix operations, multiplication, inverses Various subspaces just as setsTreat as vector spaces later(spans, column space, row space, null space, left null space) When to consider orthogonality? Vectors: orthogonal pairs, orthogonal sets, Gram-Schmidt Matrices: adjoint, Hermitian (self-adjoint), unitaryRob Beezer (U Puget Sound)A First Course in Linear AlgebraILASJune 20137 / 22
Chapter: Matrices Extended Echelon Form of m n matrix A (perhaps rectangular)RREFM [A Im ] N [B J] C0 KL Matrix on right (J) records row-operations, canonically L has rows which record “zero-ing” of rows of ARob Beezer (U Puget Sound)A First Course in Linear AlgebraILASJune 20138 / 22
Chapter: Matrices Extended Echelon Form of m n matrix A (perhaps rectangular)RREFM [A Im ] N [B J] C0 KL Matrix on right (J) records row-operations, canonically L has rows which record “zero-ing” of rows of A Then The null space of A is the null space of C ; dimension n rRob Beezer (U Puget Sound)A First Course in Linear AlgebraILASJune 20138 / 22
Chapter: Matrices Extended Echelon Form of m n matrix A (perhaps rectangular)RREFM [A Im ] N [B J] C0 KL Matrix on right (J) records row-operations, canonically L has rows which record “zero-ing” of rows of A Then The null space of A is the null space of C ; dimension n r The row space of A is the row space of C ; dimension rRob Beezer (U Puget Sound)A First Course in Linear AlgebraILASJune 20138 / 22
Chapter: Matrices Extended Echelon Form of m n matrix A (perhaps rectangular)RREFM [A Im ] N [B J] C0 KL Matrix on right (J) records row-operations, canonically L has rows which record “zero-ing” of rows of A Then The null space of A is the null space of C ; dimension n r The row space of A is the row space of C ; dimension r The column space of A is the null space of L; dimension rRob Beezer (U Puget Sound)A First Course in Linear AlgebraILASJune 20138 / 22
Chapter: Matrices Extended Echelon Form of m n matrix A (perhaps rectangular)RREFM [A Im ] N [B J] C0 KL Matrix on right (J) records row-operations, canonically L has rows which record “zero-ing” of rows of A Then The null space of A is the null space of C ; dimension n r The row space of A is the row space of C ; dimension r The column space of A is the null space of L; dimension r The left null space of A is the row space of L; dimension m rRob Beezer (U Puget Sound)A First Course in Linear AlgebraILASJune 20138 / 22
Chapters: Determinants, Eigenvalues Eigenvalues are necessarily complex numbers, even if we use Rn Ax λx then introduces vectors with complex entries So consistently work over Cn rather than Rn No penalty to do so Do not need to use complex numbers for examples Better inner product (using complex conjugation) Some theorems easier (algebraically closed field)Rob Beezer (U Puget Sound)A First Course in Linear AlgebraILASJune 20139 / 22
Chapter: Vector Spaces Have many examples of subspaces in Cn Can now formulate more axiomatic treatment Key theorem for properties of dimensionIf a set of t vectors spans the vector space V , thenany set of t 1 or more vectors is linearly dependent.Rob Beezer (U Puget Sound)A First Course in Linear AlgebraILASJune 201310 / 22
Chapter: Linear Transformations Heavy use of pre-images (a set) Parallels early theorems about solutions to systems of equations Inverse of a linear transformation Surjective: pre-images are all non-empty Injective: pre-images have at most one element Bijective: each pre-image is a singleton, so use this to establishexistence of the inverse linear transformation constructively Then exercises construct inverse linear transformations from pre-imagesof a basis of the codomainRob Beezer (U Puget Sound)A First Course in Linear AlgebraILASJune 201311 / 22
Chapter: RepresentationsVector representation is an invertible linear transformation Vector space V of dimension n with basis B {w1 , w2 , . . . , wn } ρB : V Cn ρB (v) ρB a1 a2 !n X ai w i a3 . i 1 . an Having ρ 1 is convenient (just a linear combination)Rob Beezer (U Puget Sound)A First Course in Linear AlgebraILASJune 201312 / 22
Chapter: Representations Fundamental Theorem of Matrix RepresentationT Matrix representation: MB,C(B, C bases of domain and codomain, respectively)T Then: ρC (T (u)) MB,C(ρB (u)) 1T Or: T (u) ρC MB,C(ρB (u))Rob Beezer (U Puget Sound)A First Course in Linear AlgebraILASJune 201313 / 22
A First Course in Linear Algebra, Online Used at 25 or more universities(Berkeley, Carnegie Mellon, Notre Dame, Amherst, Smith) Over 1.1 million page views at linear.pugetsound.edu #5 result in Google search for “linear algebra” Worldwide audience Version 0.10, July 2004, PDFsRob Beezer (U Puget Sound)A First Course in Linear AlgebraILASJune 201314 / 22
A First Course in Linear Algebra, Online Used at 25 or more universities(Berkeley, Carnegie Mellon, Notre Dame, Amherst, Smith) Over 1.1 million page views at linear.pugetsound.edu #5 result in Google search for “linear algebra” Worldwide audience Version 0.10, July 2004, PDFs Version 1.00, December 2006, TeX4ht conversion to HTMLRob Beezer (U Puget Sound)A First Course in Linear AlgebraILASJune 201314 / 22
A First Course in Linear Algebra, Online Used at 25 or more universities(Berkeley, Carnegie Mellon, Notre Dame, Amherst, Smith) Over 1.1 million page views at linear.pugetsound.edu #5 result in Google search for “linear algebra” Worldwide audience Version 0.10, July 2004, PDFs Version 1.00, December 2006, TeX4ht conversion to HTMLRob Beezer (U Puget Sound)A First Course in Linear AlgebraILASJune 201314 / 22
Worldwide AudienceMost recent visitors to book content, last weekend(09:51:55 29 May to 10:03:33 1 Jun, 2013)Rob Beezer (U Puget Sound)A First Course in Linear AlgebraILASJune 201315 / 22
A First Course in Linear Algebra, OnlineVersion 3.00, December 2012 Source converted to XML Web version optimized for online viewing Standard XHTML, CSS, JavaScript (“platform-independent”) Heavy cross-referencing Increased navigational aids Knowls: theorems, proofs, examples, exercises Sage cells: embedded, editable, computational examplesRob Beezer (U Puget Sound)A First Course in Linear AlgebraILASJune 201316 / 22
DemonstrationTEXTBOOK DEMOlinear.ups.edu, left sidebar: “Online”Rob Beezer (U Puget Sound)A First Course in Linear AlgebraILASJune 201317 / 22
XML SourceSection NM, Nonsingular MatricesTheorem NMRRI, Nonsingular Matrices Row-reduce to the Identity MatrixRob Beezer (U Puget Sound)A First Course in Linear AlgebraILASJune 201318 / 22
Teaching with an Online Text Nothing beats teaching with your textRob Beezer (U Puget Sound)A First Course in Linear AlgebraILASJune 201319 / 22
Teaching with an Online Text Nothing beats teaching with your text Large examples can be discussed in classRob Beezer (U Puget Sound)A First Course in Linear AlgebraILASJune 201319 / 22
Teaching with an Online Text Nothing beats teaching with your text Large examples can be discussed in class More time to discuss computational toolsRob Beezer (U Puget Sound)A First Course in Linear AlgebraILASJune 201319 / 22
Teaching with an Online Text Nothing beats teaching with your text Large examples can be discussed in class More time to discuss computational tools Many solutions includedRob Beezer (U Puget Sound)A First Course in Linear AlgebraILASJune 201319 / 22
Teaching with an Online Text Nothing beats teaching with your text Large examples can be discussed in class More time to discuss computational tools Many solutions included Students use laptops during examsOnly a simple web page with Sage cell server instancesRob Beezer (U Puget Sound)A First Course in Linear AlgebraILASJune 201319 / 22
Teaching with an Online Text Nothing beats teaching with your text Large examples can be discussed in class More time to discuss computational tools Many solutions included Students use laptops during examsOnly a simple web page with Sage cell server instances Students are involved in correcting the textRob Beezer (U Puget Sound)A First Course in Linear AlgebraILASJune 201319 / 22
Teaching with an Online Text Nothing beats teaching with your text Large examples can be discussed in class More time to discuss computational tools Many solutions included Students use laptops during examsOnly a simple web page with Sage cell server instances Students are involved in correcting the text But: our students expect a second lookRob Beezer (U Puget Sound)A First Course in Linear AlgebraILASJune 201319 / 22
Textbooks Textbook as “knowledge transfer”Rob Beezer (U Puget Sound)A First Course in Linear AlgebraILASJune 201320 / 22
Textbooks Textbook as “knowledge transfer” Much of this is happening online now,especially for 20-somethingsRob Beezer (U Puget Sound)A First Course in Linear AlgebraILASJune 201320 / 22
Textbooks Textbook as “knowledge transfer” Much of this is happening online now,especially for 20-somethings Any good textbook is available as a free downloadRob Beezer (U Puget Sound)A First Course in Linear AlgebraILASJune 201320 / 22
Textbooks Textbook as “knowledge transfer” Much of this is happening online now,especially for 20-somethings Any good textbook is available as a free download Knowls greatly improve structure of an online textbookRob Beezer (U Puget Sound)A First Course in Linear AlgebraILASJune 201320 / 22
Textbooks Textbook as “knowledge transfer” Much of this is happening online now,especially for 20-somethings Any good textbook is available as a free download Knowls greatly improve structure of an online textbook Sage, plus Sage cell server, allow integrating computationRob Beezer (U Puget Sound)A First Course in Linear AlgebraILASJune 201320 / 22
What’s Next More “products” from the source (e.g. solutions manual)Rob Beezer (U Puget Sound)A First Course in Linear AlgebraILASJune 201321 / 22
What’s Next More “products” from the source (e.g. solutions manual) A Second Course in Linear Algebra Available now:http://linear.ups.edu/version3/scla/scla.html Cross-volume linking and knowls from FCLA Numerical work with Sage (not just exact linear algebra) Completed: AY 2013-14Rob Beezer (U Puget Sound)A First Course in Linear AlgebraILASJune 201321 / 22
What’s Next More “products” from the source (e.g. solutions manual) A Second Course in Linear Algebra Available now:http://linear.ups.edu/version3/scla/scla.html Cross-volume linking and knowls from FCLA Numerical work with Sage (not just exact linear algebra) Completed: AY 2013-14 A usable system to author textbooks in XML (this summer)Rob Beezer (U Puget Sound)A First Course in Linear AlgebraILASJune 201321 / 22
FCLA: http://linear.pugetsound.eduWeb: http://buzzard.pugetsound.edu/talks.htmlBlog: http://beezers.org/blog/bbRob Beezer (U Puget Sound)A First Course in Linear AlgebraILASJune 201322 / 22
A First Course in Linear Algebra, Online Used at 25 or more universities (Berkeley, Carnegie Mellon, Notre Dame, Amherst, Smith) Over 1.1 million page views at linear.pugetsound.edu #5 result in Google search for \linear algebra" Worldwide audience Version 0.10, July 2004, PD
Robert Gerver, Ph.D. North Shore High School 450 Glen Cove Avenue Glen Head, NY 11545 gerverr@northshoreschools.org Rob has been teaching at . Algebra 1 Financial Algebra Geometry Algebra 2 Algebra 1 Geometry Financial Algebra Algebra 2 Algebra 1 Geometry Algebra 2 Financial Algebra ! Concurrently with Geometry, Algebra 2, or Precalculus
INTRODUCTION TO LINEAR ALGEBRA AND S-LINEAR ALGEBRA 1.1 Basic properties of linear algebra 7 1.2 Introduction to s-linear algebra 15 1.3 Some aapplications of S-linear algebra 30 Chapter Two INTRODUCTORY COCEPTS OF BASIC BISTRUCTURES AND S-BISTRUCTU
MTH 210: Intro to Linear Algebra Fall 2019 Course Notes Drew Armstrong Linear algebra is the common denominator of modern mathematics. From the most pure to the most applied, if you use mathematics then you will use linear algebra. It is also a relatively new subject. Linear algebra as we
High-level description of course goals: 1. linear algebra theory; 2. linear algebra computa-tional skills; 3. introduction to abstract math. Today’s topic: introduction to linear algebra. Conceptually, linear algebra is about sets of quantities (a.k.a. vectors
Sep 07, 2020 · 06 - Linear Algebra Review De ning Matrices Basic Matrix Operations Special Types of Matrices Matrix Inversion Properties of Matrices Operations of Matrices Simple Linear Regression References OverviewI We wrap up the math topics by reviewing some linear algebra concepts Linear algebra
results- rst approach to machine learning, and linear algebra is not the rst step, but perhaps the second or third. Practitioners Study Too Much Linear Algebra When practitioners do circle back to study linear algebra, they learn far more of the eld than is required for or relevant to machine learning. Linear algebra is a large eld of study
Algebra I – Advanced Linear Algebra (MA251) Lecture Notes Derek Holt and Dmitriy Rumynin year 2009 (revised at the end) Contents 1 Review of Some Linear Algebra 3 1.1 The matrix of a linear m
MA251 Algebra I: Advanced Linear Algebra 1 1 Change of Basis A major theme in MA106 Linear Algebra is change of bases. Since this is fundamental to what follows, we recall some notation and the key theorem here. Let T: U!V be a linear map between Uand V. To express Tas a matrix requires pick
