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2250 7:30am Lectures Week 7 S2013

Last Modified: February 20, 2013, 06:09 MST.    Today: October 23, 2017, 11:10 MDT.
 Edwards-Penney, sections 4.4, 4.5, 4.6, 4.7, 5.1
  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)
Edwards-Penney 5.1, 5.2, 5.3, 5.4 (15.6 K, txt, 23 Dec 2012)

Week 7, Sections 4.4, 4.5, 4.6, 4.7, 5.1

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

Tuesday: Pivot Theorem. Independence Tests. Basis and Dimension. Sections 4.4, 4.5, 4.6

 Main Results 

 THEOREM: Pivot columns are independent and non-pivot columns are
          linear combinations of the pivot columns.
 THEOREM: rank(A)=rank(A^T).
 DEF. DIMENSION.
   THEOREM. Two independent sets, each of which span a subspace S of a
            vector space V, must have the same number of vectors. This
            unique number is called the DIMENSION of the subspace S.
Slides: The pivot theorem, dimension, row rank = col rank (189.2 K, pdf, 03 Mar 2012) THEOREM: A set of nonzero pairwise orthogonal vectors is linearly independent.
Slides: Orthogonality, CSB-inequality, Pythagorean identity (124.8 K, pdf, 03 Mar 2012) BASIS. Purpose of basis: General solution with a minimal number of terms. Definition: Basis == independence + span. Differential Equations: General solution and shortest answer. Examples: Find a basis from a general solution formula. The role of partial derivatives in extracting information 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? Repeat for y = c1 + c2 exp(x). Basis calculations. Last Frame Algorithm: Basis for a linear system AX=0. Differentiate X on t1, t2, ... to find the basis. Pivot Theorem applications. The solution space of AX=0 is a subspace S (kernel theorem). S=span(columns of A)=span(pivot columns of A) Subspace basis from a spanning set. Apply the pivot theorem to find a largest set of independent vectors, generating a basis. Redundant vectors. Apply the pivot theorem to expunge irrelevant information and simplify representation of an answer. Example. Last frame algorithm and the vector general solution. Example. Given general solution y = c1 exp(x) + c2 exp(-x), What's the basis? From fixed vectors and matrices we have learned that: dim=number of partial derivatives on invented symbols basis=answers to partials on invented symbols We guess c1, c2 to be the invented symbols. Then exp(x) and exp(-x) is the basis. Independence delayed. Example. The solution space W of y'' - y = 0 has dimension 2. S1={exp(x), exp(-x)} and S2={cosh(x), sinh(x)} are two independent sets in W. By the theory, W=span(S1)=span(S2). This information simply means that there are two equally valid general solution formulas: y = c1 exp(x) + c2 exp(-x), and y = d1 cosh(x) + d2 sinh(x) FACT. Every nontrivial subspace has an infinite number of bases. PIVOT THEOREM PROOF. [slides] The pivot theorem is Algorithm 2, section 4.5.
Slides: The pivot theorem, dimension, row rank = col rank (189.2 K, pdf, 03 Mar 2012) FUNCTIONS are VECTORS How to represent functions as graphs and as infinitely long column vectors.
Slides: Functions as infinitely long column vectors (123.8 K, pdf, 22 Dec 2012) Function rules for add and scalar multiply. Independence tests using functions as the vectors. Sampling Test Wronskian Test Web References:
Slides: Vector space, subspace, independence (185.0 K, pdf, 21 Dec 2012)
Slides: The pivot theorem, dimension, row rank = col rank (189.2 K, pdf, 03 Mar 2012)
Slides: Orthogonality, CSB-inequality, Pythagorean identity (124.8 K, pdf, 03 Mar 2012)
Manuscript: Vector space, Independence, Basis, Dimension, Rank (268.7 K, pdf, 22 Mar 2010)

Wednesday: Linear Algebra Toolkit for Differential Equations. Sections 4.7, 5.1

 PARTIAL FRACTION THEORY.
      DEF. Partial fraction = constant/poly with one root
      Failsafe Method of Sampling.
        Example: 1/(x^2+3x+2)=A/(x+1) + B/(x+2)
        Example: top=x-1, bottom=(x+1)(x^2+1)
        Example: top=x-1, bottom=(x+1)^2(x^2+1)^2
      Maple assist with convert(top/bottom,parfrac,x);
      The method of atoms.
        Example: 1/(x^2+3x+2)=A/(x+1) + B/(x+2)
      Heaviside's method.
        Example: 1/(x^2+3x+2)=A/(x+1) + B/(x+2)
      BOOT CAMP
        Your job is to dig trenches and then fill them up again.
        Briefly, do all the examples above by yourself, then throw
        away the scratch paper. Become a partial fraction expert,
        better than electrical engineer Oliver Heaviside.
 PROBLEM 4.7-26.
    How to solve y''+10y'=0 for general solution y=c1 + c2 exp(-10x)
    Tools: method of quadrature and integrating factors.
 EULER 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) for real a and positive real b
  THEOREM. Euler solution atoms are independent.
  EXAMPLE. Show 1, x^2, x^9 are independent by 3 methods.
  EXAMPLE. Independence of 1, x^2, x^(3/2) by the Wronskian test.
 REVISIT PROBLEM 4.7-26.
    Equation y''+10y'=0 has general solution y=c1 + c2 exp(-10x)
    The Euler solution atoms for this example are 1 and exp(-10x).
    Differential equations like this one have general solution a
    linear combination of atoms.
 TOOLKIT for SOLVING LINEAR CONSTANT DIFFERENTIAL EQUATIONS
     Picard: Order n of a DE = dimension of the solution space.
     General solution = linear combination n independent atoms.
     Euler's theorem(s), an algorithm for solution atoms.
 ROW RANK=COL RANK PROOF by EXAMPLE. 
   rank(A)=rank(A^T). Theorem 3, section 4.5.
 ALGEBRAIC INDEPENDENCE TESTS: review and examples
   Sampling test.
      Application to x^2,exp(x)
   Wronskian test.
      Application to 1,x,x^2,x^3
   Orthogonal vector test.
      Example: (1,1,0),  (1,-1,0), ((0,0,1)
   Pivot theorem.
      Example: Find the independent columns in a matrix.
      Example: Find the maximum number of independent vectors in a list.
BASIS.
   Definition of basis and span.
   Examples: Find a basis from a general solution formula.
   Bases and the pivot theorem.
       Example: Find a basis for the row vectors in a matrix.
       Example: Find a basis or the column vectors in a matrix.
   Equivalence of bases.
       Example: A subspace S contains vectors v1,v2 and also vectors w1,w2.
                      When are both v1,v2 and w1,w2 bases for S?
   A computer test for equivalent bases.
Web References:

Slides: Orthogonality, CSB-inequality, Pythagorean identity (124.8 K, pdf, 03 Mar 2012)
Slides: The pivot theorem, dimension, row rank = col rank (189.2 K, pdf, 03 Mar 2012)
Slides: Matrix add, scalar multiply and matrix multiply (156.7 K, pdf, 03 Mar 2012)
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)

Thursday: Leif

Exam 1. Problems 4, 5.
  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: Intro to Linear Differential Equations, Sections 4.7, 5.1

Summary for Higher Order Differential Equations

Slides: Atoms, Euler's theorem, 7 examples (130.5 K, pdf, 26 Feb 2013)
Slides: Base atom, atom, basis for linear DE (117.6 K, pdf, 03 Mar 2012)
  EXAMPLE. The equation y'' +10y'=0. Review.
   How to solve y'' + 10y' = 0, chapter 1 methods. Midterm 1 problem 1(d).
    Idea: Let v=x'(t) to get a first order DE in v and a quadrature
          equation x'(t)=v(t). Solve the first order DE by the linear
          integrating factor method. Then insert the answer into
          x'(t)=v(t) and continue to solve for x(t) by quadrature.
          Vector space of functions: solution space of a differential
          equation.
   A basis for the solution space of y'' + 10y'=0 is {1,exp(-10x)}

  PICARDS THEOREM
    The theorem of Picard and Lindelof for y'=f(x,y) has an extension
    for systems of equations, which applies to scalar higher order
    linear differential equations.

   THEOREM [Picard]
     A homogeneous nth order linear differential equation with
     continuous coefficients has a general solution written as a linear
     combination of n independent solutions. This means that the
     solution space of the differential equation has dimension n.

   PICARD FAILURE.
     Although Picard's structure theorem does not provide an algorithm
       for construction of independent solutions, the theorems of Euler
       do that. Combined, there is an easy path to finding a basis for
       the solution space of an nth order linear differential equation.

  EULER'S THEOREM for CONSTRUCTING a SOLUTION BASIS
    It says y=exp(rx) is a solution of ay'' + by' + cy = 0 <==>
      r is a root of the characteristic equation ar^2+br+c=0.

    REAL EXPONENTIALS: If the root r is real, then the exponential is a
      real solution.

      THEOREM.
        A real root r=a (positive, negative or zero) produces one Euler
        solution atom exp(ax).

    COMPLEX EXPONENTIALS: If a nonreal root r=a+ib occurs, a complex
      number, then there is a conjugate root a-ib. The pair of roots
      produce two real solutions from EULER'S FORMULA (a trig topic):
                exp(i theta) = cos(theta) + i sin(theta)
      Details to obtain the two solutions will be delayed. The answer is

      THEOREM.
        A conjugate root pair a+ib,a-ib produces two independent Euler
        solution atoms exp(ax) cos(bx), exp(ax) sin(bx).

  HIGHER MULTIPLICITY
    For roots of the characteristic equation of multiplicity
    greater than one, there is a correction to the answer obtained in
    the two theorems above:
      Multiply the answers from the theorems by powers of x until
      the number of Euler solution atoms produced equals the
      multiplicity.
      EXAMPLE: If r=3,3,3,3,3 (multiplicity 5), then multiply exp(3x) by
               1, x, x^2, x^3, x^4 to obtain 5 Euler solution atoms.
      EXAMPLE: If r=5+3i,5+3i (multiplicity 2), then there are roots
               r=5-3i,5-3i, making 4 roots. Multiply the two Euler atoms
               exp(5x)cos(3x), exp(4x)sin(3x) by 1, x to obtain 4 Euler
               solution atoms.

    SHORTCUT: The characteristic equation can be synthetically formed
      from the differential equation ay''+by'+cy=0 by the formal
      replacement y ==> 1, y' ==> r, y'' ==> r^2.

  EULER'S SOLUTION ATOMS
     Leonhard Euler described a complete solution to finding n
     independent solutions in the special case when the coefficients are
     constant. The Euler solutions are called atoms in these
     lectures.

     THE TERM ATOM.
       The term atom abbreviates Euler solution atom of a
       linear differential equation. The main theorem says that the
       answer to a homogeneous constant coefficient linear differential
       equation of higher order is a linear combinations of atoms.

  EULER SOLUTION ATOMS for LINEAR DIFFERENTIAL EQUATIONS
    DEF. Base atoms are 1, exp(a x), cos(b x), sin(b x),
                        exp(ax)cos(bx), exp(ax)sin(bx).
    DEF: atom = x^n (base atom) for n=0,1,2,...

    THEOREM. Euler solution atoms are independent.

    THEOREM.
      Solutions of constant-coefficient homogeneous differential
      equations are linear combinations of a complete set of Euler
      solution atoms.

  EXAMPLE.
    The equation y''+10y'=0 has characteristic equation r^2+10r=0 with
    roots r=0, r=-10. Then Euler's theorem says exp(0x) and exp(-10x)
    are solutions. By vector space dimension theory, the Euler solution
    atoms 1, exp(-10x) are a basis for the solution space of the
    differential equation. Then the general solution is
          y = a linear combination of the Euler solution atoms
                      y = c1 (1) + c2 (exp(-10x)).