2250-1 7:30am Lecture Record Week 7 S2010

Last Modified: March 09, 2010, 09:08 MST.    Today: October 22, 2017, 14:59 MDT.

Week 7, Feb 22 to 26: Sections 4.2, 4.3, 4.4, 4.5, 4.7

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.
  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.

23 Feb: Independence and Dependence. Sections 4.1, 4.4, 4.5, 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).
   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 (132.6 K, pdf, 03 Feb 2010)
    Slides: The pivot theorem and applications (131.9 K, pdf, 02 Oct 2009)
    Slides: Orthogonality (87.2 K, pdf, 10 Mar 2008)
    Manuscript: Vector space, Independence, Basis, Dimension, Rank (268.7 K, pdf, 22 Mar 2010)
    Slides: Rank, nullity and elimination (111.6 K, pdf, 29 Sep 2009)
    Slides: Base atom, atom, basis for linear DE (85.4 K, pdf, 20 Oct 2009)

24 Feb: Pivot Theorem. Independence Tests. Basis and Dimension. Sections 4.4, 4.5

Additional Independence Tests
      Wronskian test.
      Orthogonal vector test.
      Pivot theorem [this lecture].
THEOREM: Pivot columns are independent and non-pivot columns
         are linear combinations of the pivot columns.
THEOREM: rank(A)=rank(A^T).
THEOREM: A set of nonzero pairwise orthogonal vectors is linearly independent.
Basis.
   General solutions with a minimal number of terms.
   Definition: Basis == independence + span.
   Differential Equations: General solution and shortest answer.
Pivot Theorem.
   Applications of the pivot theorem to find a largest set of independent vectors.
   Maximum set of independent vectors from a list.
PROOFS. [slides]
   The pivot theorem. Algorithm 2, section 4.5.
   rank(A)=rank(A^T). Theorem 3, section 4.5.
DIGITAL PHOTOS.
   Digital photos are matrices
   Photos are vectors == data packages
   Checkerboards and digital photos
   Matrix add and RGB separation, visualization
   Matrix scalar multiply, visualization
ANNOUNCEMENT:
  Problem session 4.3, 4.4, 4.7 on Wed in WEB 104 and Thu in JTB 140.
  Solutions to 4.3-18,24, 4.4-6,24 and 4.7-10,22,26.
  Please refer to the chapter 4 problem notes.
    Web References:
    Slides: Vector space, subspace, independence (132.6 K, pdf, 03 Feb 2010)
    Slides: The pivot theorem and applications (131.9 K, pdf, 02 Oct 2009)
    Slides: Orthogonality, CSB-inequality, Pythagorean identity (87.2 K, pdf, 10 Mar 2008)
    Manuscript: Vector space, Independence, Basis, Dimension, Rank (268.7 K, pdf, 22 Mar 2010)

24-25 Feb: Murphy

Exam 2 review.
Problem sessions on ch4 problems. Web 104 Wednesday and JTB 140 Thursday this week
  How to construct solutions to 4.3-18,24, 4.4-6,24 and 4.7-10,22,26.
  Questions answered on 4.3, 4.4, 4.7 problems.
  Survey of 4.3 problems.
   Illustration: How to do abstract independence arguments using vector
                 packages, without looking inside the packages.
   Applications of the rank test and determinant test.
   How to use the pivot theorem to identify independent vectors from a list.

26 Feb: Independence, basis and dimension


PROBLEMS.
   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)?
ALGEBRAIC TESTS: mostly review
   Rank test.
   Determinant test.
   Sampling test.
   Wronskian test.
   Orthogonal vector test.
   Pivot theorem.
   FUNCTIONS.
     How to represent functions as graphs and as infinitely long column
       vectors. Rules for add and scalar multiply. Independence tests
       using functions as the vectors.
BASIS.
   Definition of basis and span.
   Examples: Find a basis from a general solution formula.
   Bases and the pivot theorem.
DIMENSION.
   THEOREM. Two bases for a vector space V must have the same number of vectors.

   Examples:
     Last Frame Algorithm: Basis for a linear system Ax=0.
     Last frame algorithm and the vector general solution.
     Basis of solutions to a homogeneous system of linear algebraic equations.
     Bases and partial derivatives of the general solution on the invented symbols t1, t2, ...
     DE Example: y = c1 e^x + c2 e^{-x} is the general solution. What's the basis?
SOLUTION ATOMS and INDEPENDENCE.
  Def. atom=x^n(base atom)
       base atom = 1, exp(ax), cos(bx), sin(bx), exp(ax) cos(bx), exp(ax) sin(bx)
       "atom" abbreviates "solution atom of a linear differential equation"
  THEOREM. Atoms are independent.
  EXAMPLE. Show 1, x^2, x^9 are independent
  EXAMPLE. Show 1, x^2, x^(3/2) are independent [Wronskian test]
  PROBLEM 4.7-26.
     How to solve y''+10y'=0 for general solution y=c1 + c2 exp(-10x)
    Web References:
    Slides: Orthogonality, CSB-inequality, Pythagorean identity (87.2 K, pdf, 10 Mar 2008)
    Slides: The pivot theorem and applications (131.9 K, pdf, 02 Oct 2009)
    Slides: Matrix add, scalar multiply and matrix multiply (122.5 K, pdf, 02 Oct 2009)
    Slides: Digital photos, Maxwell's RGB separations, visualization of matrix add (153.7 K, pdf, 16 Oct 2009)
    Slides: More on digital photos, checkerboard analogy (109.5 K, pdf, 02 Oct 2009)
    Slides: Vector space, subspace, independence (132.6 K, pdf, 03 Feb 2010)
    Manuscript: Vector space, Independence, Basis, Dimension, Rank (268.7 K, pdf, 22 Mar 2010)