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2250-2 7:30am Lecture Record Week 6 F2010

Last Modified: October 03, 2010, 10:05 MDT.    Today: October 21, 2017, 11:26 MDT.

27 Sep, 29 Sep, 30 Sep: Michal and Laura


Exam 1. Problems 4,5. Take the exam 27 Sep or 29 Sep in JWB 335 at 12:50pm, or 30 Sep in WEB 103 at 7:25am.
Sample Exam: Exam 1 keys from S2010 and F2009. See also S2009, exam 1.
HTML: Exam links for the past 5 years (17.7 K, html, 18 Dec 2010)

27 Sep: Determinants, Survey of Results. Section 3.6

More theorems on inverses
THEOREM. The inverse of inverse(A) is A itself.
THEOREM. If C and D have inverses, then so does CD and
                      inverse(CD) = inverse(D) inverse(C).
College Algebra Background:
   College algebra determinant definition
   Sarrus' rule for 2x2 and 3x3 matrices.
    References for 3.6 determinant theory and Cramer's Rule
    Slides: Determinants 2008 (167.7 K, pdf, 23 Sep 2010)
    Manuscript: Determinants, Cramer's rule, Cayley-Hamilton (186.5 K, pdf, 09 Aug 2009)
Methods for computing a determinant
  Sarrus' rule, 2x2 and 3x3 cases.
  Four rules for determinants
     Triangular Rule (one-arrow Sarrus' Rule): The determinant of
        a triangular matrix is the product of the diagonal elements.
     Multiply rule: B=answer after mult(t,m), then |A| = (1/m) |B|
     Swap rule: B=answer after swap(s,t), then |A| = (-1) |B|
     Combo rule: B=answer after combo(s,t,c), then |A| = |B|
Results on Determinants
     Examples: Computing det(A) easily. When does det(A)=0?
    THEOREM. Determinant values for elementary matrices:
        det(E)=1 for combo(s,t,c),
        det(E)=m for mult(t,m),
        det(E)=-1 for swap(s,t).
Survey of Main theorems:
  Computation by the 4 rules, cofactor expansion, hybrid methods.
  Determinant product theorem det(AB)=det(A)det(B).
  Cramer's Rule for solving Ax=b:
   
x1 = delta1/delta, ... , xn = deltan/delta Adjugate formula: A adj(A) = adj(A) A = det(A) I Adjugate inverse formula inverse(A) = adjugate(A)/det(A).

28 Sep: Cramers Rule, Adjugate formula. Section 3.6

Lecture
 Cofactor 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.
   How to form the matrix of cofactors and its transpose.
   The adjugate matrix.
 
THEOREM. The 4 rules for computing any determinant can be compressed into two rules,
  1. det(triangular matrix)=the product of the diagonal elements, and
  2. det(EA)=det(E)det(A), where E is an elementary combo, swap or mult matrix.
Determinant product theorem
    det(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).
Slides: Determinants 2010 (167.7 K, pdf, 23 Sep 2010)
Manuscript: Determinants, Cramer's rule, Cayley-Hamilton (186.5 K, pdf, 09 Aug 2009)
html: Problem notes F2010 (4.6 K, html, 26 Nov 2010)

29 Sep: Problem session 3.4, 3.5, 3.6

       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 added more details, and complete solutions in several cases.

   Maple computation of det(A), invverse(A), adjoint(A)

1 Oct: Problem Session Ch 3. Introduction to Chapter 4. Vector Space. Section 4.1.

Transpose matrix
        (A^T)^T = A
        (A + B)^T = A^T + B^T
        (AB)^T = B^T A^T
Further properties of the adjugate matrix
   Computing 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
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 (186.5 K, pdf, 09 Aug 2009)
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. How to solve problem 3.4-30. 3.5-44 proof, revisited.
 Intro to Ch4
  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.

References for chapters 3 and 4, Linear Algebra


    Manuscript: Linear algebraic equations, no matrices (292.8 K, pdf, 01 Feb 2010)
    Slides: vector models and vector spaces (110.3 K, pdf, 03 Oct 2009)
    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 (266.8 K, pdf, 09 Aug 2009)
    Manuscript: Matrix Equations (162.6 K, pdf, 09 Aug 2009)
    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 (114.4 K, pdf, 03 Oct 2009)
    Slides: Elementary matrices, vector spaces (35.8 K, pdf, 18 Feb 2007)
    Slides: Three possibilities, theorems on infinitely many solutions, equations with symbols (101.0 K, pdf, 28 Sep 2010)
    Beamer slides: 3 possibilities with symbol k (60.0 K, pdf, 31 Jan 2010)
    Slides: 3 possibilities with symbol k (72.8 K, pdf, 31 Jan 2010)
    Slides: Linear equations, reduced echelon, three rules (155.6 K, pdf, 06 Aug 2009)
    Slides: Infinitely many solutions case (93.8 K, pdf, 03 Oct 2009)
    Slides: No solution case (58.4 K, pdf, 03 Oct 2009)
    Slides: Unique solution case (86.0 K, pdf, 03 Oct 2009)
    Maple: Lab 5, Linear algebra (170.1 K, pdf, 17 Aug 2010)
    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 F2010 (4.6 K, html, 26 Nov 2010)
    Slides: Determinants 2010 (167.7 K, pdf, 23 Sep 2010)
    Manuscript: Determinants, Cramers rule, Cayley-Hamilton (186.5 K, pdf, 09 Aug 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: Inverse matrix, frame sequence method (71.6 K, pdf, 02 Oct 2009)
    Slides: More on digital photos, checkerboard analogy (109.5 K, pdf, 02 Oct 2009)
    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)
    Slides: Orthogonality (78.9 K, pdf, 14 Oct 2010)
    Slides: Partial fraction theory (121.5 K, pdf, 30 Aug 2009)
    Slides: The pivot theorem and applications (132.5 K, pdf, 10 Oct 2010)
    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)