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2250-1 7:30am Lecture Record Week 7 S2012

Last Modified: February 25, 2012, 20:25 MST.    Today: December 11, 2017, 16:03 MST.

Week 7, Feb 20 to 24: Sections 4.1, 4.2, 4.3, 4.4, 4.5

This is a 4-day week with only three lectures and one exam day. Monday was a holiday, President's Day.

Tue, 21 Feb: Problem Session Ch 3. Introduction to Chapter 4. Vector Space. Section 4.1.

REVIEW PROBLEMS Ch3
   3.6-60: Reading on induction. Required details.
      B_n = 2B_{n-1} - B_{n-2},  B_n = n+1
   3.6-review: matrix A is 10x10 and has 92 ones. What's det(A)?
       Problem 3.5-60a and 60b.
          How to discover the relation B_n = 2 B_{n-1} - B_{n-2}
          Induction proof in 3.5-60b.

       Problems 3.3-10,20 using maple
       Problems 3.4-20,30,34,40
       Problems 3.5-16,26,44
       Problems 3.6-6,20,32,40,60
   Some problem details appear already in the online problem notes.
   The lecture adds more details, and complete solutions in several cases.

   Maple computation of det(A), inverse(A), adjoint(A)
Problem Details
  Exercises 3.4-34 and 3.4-40.
  Cayley-Hamilton Theorem.
    It is a famous result in linear algebra which is the basis for
    solving systems of differential equations.
    Discussion of the Cayley-Hamilton theorem [Exercise 3.4-29;
      see also Section 6.3]

Manuscript: Determinants, Cramer's rule, Cayley-Hamilton (326.0 K, pdf, 07 Feb 2012)
Superposition proof Problem 3.4-40 is the superposition principle for the matrix equation Ax=b. It is the analog of the differential equation relation y=y_h + y_p. Web notes on the problems. Problem 3.4-29 is used in Problem 3.4-30. See FAQ for 3.4. Explained there: How to solve problem 3.4-30. For the 3.5-44 proof, see the 3.5 FAQ.
Four Vector Models:
   Fixed vectors
   Triad i,j,k algebraic calculus model
   Physics and Engineering arrows
   Gibbs vectors.

Slides: vector models and vector spaces (144.1 K, pdf, 03 Mar 2012) Parallelogram law. Head minus tail rule. Vector Toolkit The 8-property toolkit for vectors. Vector spaces. Reading: Section 4.1 in Edwards-Penney, especially the 8 properties. Lecture: Abstract vector spaces. Def: Vector==package of data items. Vectors are not arrows. The 8-Property Vector Toolkit Def: vector space, subspace Working set == subspace. Data set == Vector space Examples of vectors: Digital photos, Fourier coefficients, Taylor coefficients, Solutions to DE. Example: y=2exp(-x^2) for DE y'=-2xy, y(0)=2. 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)

22 Feb: Subspace Tests and Applications. Sections 4.2, 4.3.

  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.
  More on vector spaces and subspaces:
    Detection of subspaces and data sets that are not subspaces.
    Theorems:
       Subspace criterion,
       Kernel theorem,
       Not a subspace theorem.
       The Span Theorem.
  Use of theorems 1,2 in section 4.2.
  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 (thm 2 page 239 or 243
     section 4.2 of Edwards-Penney).
  Subspace applications.
    When to use the kernel theorem.
    When to use the subspace criterion.
    When to use the not a subspace theorem.
    Problems 4.1,4.2.
Textbook: Chapter 4, sections 4.1 and 4.2.
Web references for chapter 4. Repeated below in ch3-ch4 references.
Slides: Vector space, subspace, independence (182.3 K, pdf, 03 Mar 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 (154.1 K, pdf, 03 Mar 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 S2012 (5.2 K, html, 08 Apr 2012)

24 Feb: Independence and Dependence. Sections 4.1, 4.4, 4.7

Lecture: Sections 4.1, 4.4 and some part of 4.7.
Drill:
  The 8-property vector toolkit.
    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.
Review of Vector spaces.
  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.
Lecture: 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.
 Algebraic tests.
   Rank test.
   Determinant test.
   Sampling test.
   Additional tests
      Wronskian test.
      Orthogonal vector test.
      Pivot theorem.
 Geometric tests.
   One or two vector independence.
   Geometry of dependence in dimensions 1,2,3.
    Web References:
    Slides: Vector space, subspace, independence (182.3 K, pdf, 03 Mar 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 (154.1 K, pdf, 03 Mar 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 (869.6 K, pdf, 25 Sep 2003)
    Transparencies: Ch4 all, Exercises 4.1 to 4.7 (461.2 K, pdf, 03 Oct 2010)
    Transparency: Sample solution ER-1 [same as L3.1] (184.6 K, jpg, 08 Feb 2008)
    Slides: Elementary matrix theorems (161.5 K, pdf, 03 Mar 2012)
    Slides: Elementary matrices, vector spaces (35.8 K, pdf, 18 Feb 2007)
    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 (233.6 K, pdf, 03 Mar 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 (117.8 K, pdf, 03 Mar 2012)
    Maple: Lab 5, Linear algebra (71.4 K, pdf, 09 Dec 2011)
    Slides: Three rules, frame sequence, maple syntax (35.8 K, pdf, 25 Jan 2007)
    Transparencies: 3x3 Frame sequence and general solution (90.0 K, pdf, 28 Sep 2006)
    html: Problem notes S2012 (5.2 K, html, 08 Apr 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 (154.1 K, pdf, 03 Mar 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)
    Text: Lawrence Page's pagerank algorithm (0.7 K, txt, 06 Oct 2008)
    Text: History of telecom companies (1.4 K, txt, 30 Dec 2009)