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2250 7:30am Lectures Week 6 S2013

Last Modified: February 12, 2013, 05:45 MST.    Today: October 21, 2017, 09:58 MDT.
 Edwards-Penney, sections 4.1, 4.2, 4.3, 4.4
  The textbook topics, definitions and theorems
Edwards-Penney 4.1, 4.2, 4.3, 4.4 (5.6 K, txt, 24 Dec 2012)
Edwards-Penney 4.5, 4.6, 4.7 (7.0 K, txt, 24 Dec 2012)

Monday: Problem Session 3.3, 3.4, 3.5, 3.6

New Topics
  Rank, Nullity, Dimension and Elimination for Equations
      Three possibilities
      Definitions: rank, nullity, dimension
      Rank-Nullity theorem
      Elimination algorithm
      Examples
  References:
Slides: Rank, nullity and elimination (156.3 K, pdf, 21 Dec 2012) Ch 3 PROBLEMS Problems 3.3-10,20 using maple Problem 3.4-20 Long details in FAQ 3.4 Problem 3.4-30, Cayley-Hamilton Problem 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
Slides: Cayley-Hamilton Theorem (100.5 K, pdf, 21 Dec 2012)
Manuscript: Determinants, Cramer's rule, Cayley-Hamilton (326.0 K, pdf, 07 Feb 2012) Problem 3.4-40, Superposition proof The 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 induction

Tuesday: Vector spaces R^3 and R^n. Sections 4.1, 4.2.

Intro to Ch4
  Def: Vector==package of data items
  Vector Toolkit
     The 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:
Slides: vector models and vector spaces (144.1 K, pdf, 03 Mar 2012) Parallelogram law. Head minus tail rule. Abstract 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.
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)

Wednesday: Subspace Tests and Applications. Sections 4.2, 4.3.

  Subspaces
    Data recorder example
      A 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
Slides: Vector space, subspace, independence (185.0 K, pdf, 21 Dec 2012) 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 references
Slides: Vector space, subspace theorems, independence tests (185.0 K, pdf, 21 Dec 2012)
Slides: Orthogonal vector theorem (124.8 K, pdf, 03 Mar 2012) 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 (185.0 K, pdf, 21 Dec 2012)
Manuscript: Vector space, Independence, Basis, Dimension, Rank (268.7 K, pdf, 22 Mar 2010)
Slides: The pivot theorem and applications (189.2 K, pdf, 03 Mar 2012)
Slides: Rank, nullity and elimination (156.3 K, pdf, 21 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, 03 Mar 2012)
Transparencies: Ch4 Page 237+ slides, Exercises 4.1 to 4.4, some 4.9 (463.2 K, pdf, 25 Sep 2003)
html: Problem notes S2013 (5.1 K, html, 06 Dec 2012)

Thursday: Leif

Exam 1. Problems 1, 2, 3.
  Sample Exam: Exam 1 keys from S2012 and F2010.
HTML: Exam links for the past 5 years (19.6 K, html, 30 Apr 2013)

Friday: Independence and Dependence. Sections 4.1, 4.3, 4.7

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.
Quick Review of Vector spaces 4.1, 4.2
  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.
Web References
Slides: Vector space, subspace, independence tests (185.0 K, pdf, 21 Dec 2012)
Slides: The pivot theorem and applications (189.2 K, pdf, 03 Mar 2012)
Slides: Orthogonality (124.8 K, pdf, 03 Mar 2012)
Manuscript: Vector space, Independence, Basis, Dimension, Rank (268.7 K, pdf, 22 Mar 2010)
Slides: Rank, nullity and elimination (156.3 K, pdf, 21 Dec 2012)
Slides: Base atom, atom, basis for linear DE (117.6 K, pdf, 03 Mar 2012)

References for chapters 3 and 4, Linear Algebra


Manuscript: Linear algebraic equations, no matrices (460.2 K, pdf, 10 Feb 2012)
Slides: vector models and vector spaces (144.1 K, pdf, 03 Mar 2012)
Manuscript: 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)
Transparencies: 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)
Transparency: Sample solution ER-1 [exam review and maple lab 3] (184.6 K, jpg, 08 Feb 2008)
Slides: Elementary matrix theorems (161.5 K, pdf, 03 Mar 2012)
Slides: Three possibilities, theorems on infinitely many solutions, equations with symbols (129.0 K, pdf, 03 Mar 2012)
Beamer slides: 3 possibilities with symbol k (60.0 K, pdf, 31 Jan 2010)
Slides: 3 possibilities with symbol k (98.8 K, pdf, 03 Mar 2012)
Slides: Linear equations, reduced echelon, three rules (237.3 K, pdf, 15 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, 14 Dec 2012)
Slides: Three rules, frame sequence, maple syntax (35.8 K, pdf, 25 Jan 2007)
Jpeg: 3x3 Frame sequence and general solution (315.9 K, jpg, 12 Dec 2012)
Slides: Determinants 2012 (227.1 K, pdf, 03 Mar 2012)
Manuscript: Determinants, Cramers rule, Cayley-Hamilton (326.0 K, pdf, 07 Feb 2012)
Slides: 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 (97.9 K, pdf, 03 Mar 2012)
Slides: More on digital photos, checkerboard analogy (141.9 K, pdf, 03 Mar 2012)
Slides: Rank, nullity and elimination (156.3 K, pdf, 21 Dec 2012)
Slides: Base atom, atom, basis for linear DE (117.6 K, pdf, 03 Mar 2012)
Slides: Orthogonality (124.8 K, pdf, 03 Mar 2012)
Slides: Partial fraction theory (160.7 K, pdf, 03 Mar 2012)
Slides: The pivot theorem and applications (189.2 K, pdf, 03 Mar 2012)
Maple: Lab 5, Linear algebra (170.0 K, pdf, 04 Dec 2012)
html: Problem notes S2013 (5.1 K, html, 06 Dec 2012)
Text: Lawrence Page's pagerank algorithm (0.7 K, txt, 06 Oct 2008)
Text: History of telecom companies (1.9 K, txt, 04 Apr 2013)