Edwards-Penney, sections 4.1, 4.2, 4.3, 4.4 The textbook topics, definitions and theoremsThis is a 4-day week with Monday holiday.

Edwards-Penney 4.1, 4.2, 4.3, 4.4 (5.6 K, txt, 18 Dec 2013)

Edwards-Penney 4.5, 4.6, 4.7 (7.0 K, txt, 18 Dec 2013)

Discussion of 3.5 problems. Lecture Ideas of rank, nullity, dimension in examples.: Rank, nullity and elimination (156.3 K, pdf, 20 Dec 2012)SlidesLecture on Cofactor MethodCofactor expansion of det(A). minor(A,i,j) checkerboard sign (-1)^{i+j} cofactor(A,i,j)=(sign)minor(A,i,j) Details for 3x3 and 4x4. Hybrid methods to evaluate det(A). How to use the 4 rules to compute det(A) for any size matrix. Computing determinants of sizes 3x3, 4x4, 5x5 and higher. Frame sequences and determinants. Formula for det(A) in terms of swap and mult operations. Special theorems for det(A)=0 a zero row or col duplicates rows proportional rows. Elementary matrices Determinant product rule for an elementary matrix Cramer's rule. Adjugate Matrix. How to form the matrix of cofactors and its transpose. DEF: The adjugate matrix.THEOREM.The 4 rules for computing any determinant can be compressed into two rules, det(triangular matrix)=the product of the diagonal elements, det(EA)=det(E)det(A), where E is an elementary matrix, combo, swap or mult.Determinant product theoremdet(AB)=det(A)det(B) for any two square matrices A,B Proof details. Example.THEOREM. The adjugate formula A adj(A) = adj(A) A = det(A) I.THEOREM. Adjugate inverse formula: inverse(A) = adj(A)/det(A).: Determinants 2010 (227.1 K, pdf, 03 Mar 2012)Slides: Determinants, Cramer's rule, Cayley-Hamilton (326.0 K, pdf, 07 Feb 2012)Manuscript: Problem notes S2015 (2.8 K, html, 01 Jan 2015)htmlREVIEW: Transpose matrix(A^T)^T = A (A + B)^T = A^T + B^T (AB)^T = B^T A^T det(A)=det(A^T)Further properties of the adjugate matrixComputing det(A) from A and adj(A) in 10 seconds Problems involving adj(A): examples from exams. Adjugate identity A adj(A) = adj(A) A = det(A) I 3x3 case: 6 ways to compute det(A) from A, adj(A). 3x3 case: the 6 cofactor expansions

New TopicsRank, Nullity, Dimension and Elimination for EquationsThree possibilities Definitions: rank, nullity, dimension Rank-Nullity theorem Elimination algorithm Examples References:: Rank, nullity and elimination (156.3 K, pdf, 20 Dec 2012)SlidesCh 3 PROBLEMSProblems 3.3-10,20 using maple Problem 3.4-20 Long details in FAQ 3.4 Problem 3.4-30,Cayley-HamiltonProblem 3.4-29 is used in Problem 3.4-30. See FAQ 3.4 for details Cayley-Hamilton Theorem. It is a famous result in linear algebra which is the basis for solving systems of differential equations. Discussion: Cayley-Hamilton Theorem (100.5 K, pdf, 20 Dec 2012)Slides: Determinants, Cramer's rule, Cayley-Hamilton (326.0 K, pdf, 07 Feb 2012)Manuscript

ExercisesProblem 3.4-40,Superposition proofThe problem is to prove the superposition principle for the matrix equation Ax=b. It is the analog of the differential equation relation y=y_h + y_p. Details in FAQ 3.4. Problems 3.5-16,26,44 For the 3.5-44 proof, see the 3.5 FAQ. Problems 3.6-6,20,32,40,60 3.6 FAQ for details and answer checks Maple Answer Checks: Compute det(A), inverse(A), adjoint(A) Review 3.6. matrix A is 10x10 and has 92 ones. What's det(A)? Problem 3.6-60, nxn determinants (60a) B_n = 2B_{n-1} - B_{n-2}, by cofactor expansion (60b) B_n = n+1 by inductionIntro to Ch4Def: Vector==package of data itemsVector ToolkitThe 8-property toolkit for vectors [4.2] Reading: Sections 4.1, 4.2 in Edwards-Penney Def: vector space, subspace Data set == Vector space Working set == subspace. Examples of vectors: Four classical vector models, Vectors are not arrows Fixed vectors Triad i,j,k algebraic calculus model Physics and Engineering arrows Gibbs vectors. Digital photos, Fourier coefficients, Taylor coefficients, Solutions to DE. Example: y=2exp(-x^2) for DE y'=-2xy, y(0)=2.Four Vector Models:: vector models and vector spaces (144.1 K, pdf, 03 Mar 2012) Parallelogram law. Head minus tail rule.SlidesAbstract vector spaces, 4.2.Def: Vector==package of data items. Vectors are not arrows The 8-Property Vector Toolkit Def: abstract vector space Data set == Vector space Def: Subspace of a vector space Working set == smaller vector space = subspace Vector space of color photographs RGB color separation and matrix add Intensity adjustments and scalar multiply Digital photos and matrix add, scalar multiply visualization.: Digital photos, Maxwell's RGB separations, visualization of matrix add (170.2 K, pdf, 03 Mar 2012)Slides: More on digital photos, checkerboard analogy (141.9 K, pdf, 03 Mar 2012)Slides

SubspacesData recorder exampleA certain planar kinematics problem records the data set V using three components x,y,z. The working set S is a plane described by an ideal equation ax+by+cz=0. This plane is the hidden subspace of the physical application, obtained by a computation on the original data set V. Web reference: Vector space, subspace, independence (168.4 K, pdf, 17 Feb 2014) More on vector spaces and subspaces Detection of subspaces and data sets that are not subspaces. Subspace Theorems: Subspace criterion, Kernel theorem, Not a subspace theorem. The Span Theorem. Preview: Independence-Dependence Theorems Determinant test Rank test Pivot theorem Orthogonal vector theorem Wronskian test for functions Sample test for functions Web referencesSlides: Vector space, subspace theorems, independence tests (168.4 K, pdf, 17 Feb 2014)Slides: Orthogonal vector theorem (124.8 K, pdf, 04 Dec 2014) Use of subspace theorems 1,2 in section 4.2. Subspace problem types in 4.1, 4.2. Example: Subspace Shortcut for the set S in R^3 defined by x+y+z=0. Avoid using the subspace criterion on S, by writing it as Ax=0, followed by applying the kernel theorem (4.2 Theorem 2). Subspace applications. When to use the kernel theorem. When to use the subspace criterion. When to use the not a subspace theorem. Identifying a subspace with the span theorem Identifying a subspace defined by equations Problems 4.1,4.2. Textbook Reading: Chapter 4, sections 4.1 and 4.2. Web references for chapter 4.Slides: Vector space, subspace, independence (168.4 K, pdf, 17 Feb 2014)Slides: Vector space, Independence, Basis, Dimension, Rank (268.7 K, pdf, 22 Mar 2010)Manuscript: The pivot theorem and applications (189.2 K, pdf, 03 Mar 2012)Slides: Rank, nullity and elimination (156.3 K, pdf, 20 Dec 2012)Slides: Digital photos, Maxwell's RGB separations, visualization of matrix add (170.2 K, pdf, 03 Mar 2012)Slides: More on digital photos, checkerboard analogy (141.9 K, pdf, 03 Mar 2012)Slides: Orthogonality (124.8 K, pdf, 04 Dec 2014)Slides: Ch4 Page 237+ slides, Exercises 4.1 to 4.4, some 4.9, from EP 2/E (463.2 K, pdf, 25 Sep 2003)Transparencies: Problem notes (2.8 K, html, 01 Jan 2015)html

Lab 5 due.Lab 6 introduced.Sample Exam: Exam 1 and Exam 2 keys from S2014. See also S2012 and 2013, exams 1 and 2. The first exam covers chapters 1,2,3.

Sample Midterm 1, S2015, with solutions (1456.8 K, pdf, 19 Feb 2015)

Exams and exam keys for the last 5 years (22.2 K, html, 23 Feb 2015)

Sections 4.1, 4.3 and some part of 4.7. Review: Is the 8-property vector toolkit good for nothing? Example: Prove zero times a vector is the zero vector. The kernel: Solutions of Ax=0. Find the kernel of the 2x2 matrix with 1 in the upper right corner and zeros elsewhere. This is a key example in the theory of eigenanalysis.

Vectors as packages of data items. Vectors are not arrows. Examples of vector packaging in applications. Fixed vectors. Gibbs motions. Physics i,j,k vectors. Arrows in engineering force diagrams. Functions, solutions of DE. Matrices, digital photos. Sequences, coefficients of Taylor and Fourier series. Hybrid packages. The toolkit of 8 properties. Subspaces. Data recorder example. Data conversion to fit physical models. Subspace criterion (Theorem 1, 4.2). Kernel theorem (Theorem 2, 4.2). Span Theorem (Theorem 1, 4.3) Not a Subspace Theorem (Theorem 1 backwards)Independence and dependence.Example: c1 e^x+ c2 xe^{-x} = 2 e^x + 3 e^{-x} ==> c1=2, c2=3. Solutions of differential equations are vectors. Geometric tests One vector v1. Two vectors v1, v2. Three vectors v1, v2, v3. Abstract vector space tests One vector v1. Two vectors v1, v2. Algebraic tests. Rank test. Determinant test. Pivot theorem. Additional tests Sampling test. Wronskian test. Orthogonal vector test. THEOREM: Pivot columns are independent and non-pivot columns are linear combinations of the pivot columns.## References for chapters 3 and 4, Linear Algebra

: Linear algebraic equations, no matrices (429.7 K, pdf, 30 Jan 2014)Manuscript: vector models and vector spaces (144.1 K, pdf, 03 Mar 2012)Slides: Linear equations, reduced echelon, three rules (45.8 K, pdf, 22 Sep 2006)Manuscript: Three rules, frame sequence, maple syntax (35.8 K, pdf, 25 Jan 2007)Manuscript: Vectors and Matrices (426.4 K, pdf, 07 Feb 2012)Manuscript: Matrix Equations (292.1 K, pdf, 07 Feb 2012)Manuscript: Ch3 all, Exercises 3.1 to 3.6 from EP 2nd Edition (869.6 K, pdf, 25 Sep 2003)Transparencies: Ch4 all, Exercises 4.1 to 4.7, from EP 2nd edition (461.2 K, pdf, 03 Oct 2010)Transparencies: Elementary matrix theorems (154.2 K, pdf, 11 Feb 2015)Slides: Three possibilities, theorems on infinitely many solutions, equations with symbols (129.0 K, pdf, 03 Mar 2012)Slides: 3 possibilities with symbol k (60.0 K, pdf, 31 Jan 2010)Beamer slides: 3 possibilities with symbol k (98.8 K, pdf, 03 Mar 2012)Slides: Linear equations, reduced echelon, three rules (237.3 K, pdf, 14 Dec 2012)Slides: Infinitely many solutions case (131.1 K, pdf, 03 Mar 2012)Slides: No solution case (79.7 K, pdf, 03 Mar 2012)Slides: Unique solution case (110.1 K, pdf, 13 Dec 2012)Slides: Three rules, frame sequence, maple syntax (35.8 K, pdf, 25 Jan 2007)Slides: 3x3 Frame sequence and general solution (315.9 K, jpg, 12 Dec 2012)Jpeg: Determinants 2012 (227.1 K, pdf, 03 Mar 2012)Slides: Determinants, Cramers rule, Cayley-Hamilton (326.0 K, pdf, 07 Feb 2012)Manuscript: Matrix add, scalar multiply and matrix multiply (156.7 K, pdf, 03 Mar 2012)Slides: Digital photos, Maxwell's RGB separations, visualization of matrix add (170.2 K, pdf, 03 Mar 2012)Slides: Inverse matrix, frame sequence method (97.0 K, pdf, 11 Feb 2015)Slides: More on digital photos, checkerboard analogy (141.9 K, pdf, 03 Mar 2012)Slides: Rank, nullity and elimination (156.3 K, pdf, 20 Dec 2012)Slides: Base atom, atom, basis for linear DE (117.6 K, pdf, 03 Mar 2012)Slides: Orthogonality (124.8 K, pdf, 04 Dec 2014)Slides: Partial fraction theory (148.6 K, pdf, 14 Dec 2014)Slides: The pivot theorem and applications (189.2 K, pdf, 03 Mar 2012)Slides: Problem notes S2015 (2.8 K, html, 01 Jan 2015)html: Lawrence Page's pagerank algorithm (0.7 K, txt, 06 Oct 2008)Text: History of telecom companies (1.9 K, txt, 03 Apr 2013)Text