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2250-4 12:55pm Lecture Record Week 5 S2010

Last Modified: February 14, 2010, 10:28 MST.    Today: October 18, 2017, 18:09 MDT.

9 Feb: Augmented Matrix for System Ax=b. RREF. Last Frame Algorithm. Sections 3.3, 3.4.

Review
  The three possibilities
  Frame sequence analysis and the general solution.
  Frame sequences with symbol k.
  Last frame test.
  Last frame algorithm.
   Scalar form of the solution.
   Preview: Vector form of the solution.
Lecture: 3.3 and 3.4
  Translation of equation models
   Equality of vectors
   Scalar equations translate to augmented matrix
   Augmented matrix translate to scalar equations
  Matrix toolkit: Combo, swap and multiply
  Frame sequences for matrix models.
  Special matrices
    Zero matrix
    identity matrix
    diagonal matrix
    upper and lower triangular matrices
    square matrix
  THEOREM. Homogeneous system with a unique solution.
  THEOREM. Homogeneous system with more variables than equations.
     Equation ideas can be used on a matrix A.
     View matrix A as the set of coefficients of a homogeneous
     linear system Ax=0. The augmented matrix B for this homogeneous
     system would be the given matrix with a column of zeros appended:
     B=aug(A,0).
  Answer checks
     matlab, maple and mathematica.
     Pitfalls.
  Answer checks should also use the online FAQ.

html: Problem notes S2010 (4.4 K, html, 31 Jan 2010)

Last Frame Algorithm

How to use maple to compute a frame sequence. Example is Exercise 3.2-14 from Edwards-Penney.
Frame sequences with symbol k.

    Maple Worksheet: Frame Sequence in maple, Exercise 3.2-14 (3.1 K, mws, 23 Sep 2009)
    Maple Text: Frame Sequence in maple, Exercise 3.2-14 (2.8 K, txt, 23 Sep 2009)
    Slides: Three possibilities, theorems on infinitely many solutions, equations with symbols (100.3 K, pdf, 23 Sep 2009)
    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)
    Manuscript: Example 10 in Linear algebraic equations no matrices (292.8 K, pdf, 01 Feb 2010)
Matrices
    Vector.
    Matrix multiply
    The college algebra definition
    Examples.
    Matrix rules
      Vector space rules.
      Matrix multiply rules.
    Examples: how to multiply matrices on paper.

Slides: Matrix add, scalar multiply and matrix multiply (122.5 K, pdf, 02 Oct 2009)
Manuscript: Vectors and Matrices (266.8 K, pdf, 09 Aug 2009)
Manuscript: Matrix Equations (162.6 K, pdf, 09 Aug 2009)
  General structure of linear systems.
    Superposition.
    General solution
       X=X0+t1 X1 + t2 X2 + ... + tn Xn.
  Matrix formulation Ax=b of a linear system
  Properties of matrices: addition, scalar multiply.
  Matrix multiply rules. Matrix multiply Ax for x a vector.
  Linear systems as the matrix equation Ax=b.

9 Feb: Elementary matrices. Section 3.5

Lecture: Elementary matrices.
  How to write a frame sequence as a matrix product
  Fundamental theorem on frame sequences

    THEOREM. If A1 and A2 are the first two frames of a sequence,
             then A2=E A1, where E is the elementary matrix built
             from the identity matrix I by applying one toolkit
             operation combo(s,t,c), swap(s,t) or mult(t,m).

    THEOREM. If a frame sequence starts with A and ends with B, then
             B = (product of elementary matrices) A.

    The meaning: If A is the first frame and B a later frame in a
                 sequence, then there are elementary swap, combo
                 and mult matrices E1 to En such
                 that the frame sequence A ==> B can be written as
                 the matrix multiply equation
                   B=En En-1 ... E1 A.
    Web References: Elementary matrices
    Slides: vector models and vector spaces (110.3 K, pdf, 03 Oct 2009)
    Slides: Elementary matrix theorems (114.4 K, pdf, 03 Oct 2009)
    Slides: Elementary matrices, vector spaces (35.8 K, pdf, 18 Feb 2007)

11 Feb: Inverses. Rank and nullity. Section 3.5.

Discussion of 3.4 problems.
How to compute the inverse matrix 
      Def: AB=BA=I means B is the inverse of A.
      Inverse = adjugate/determinant (2x2 case)
      Frame sequences method.
      Inverse rules
      Web References: Construction of inverses. Theorems on inverses.

      THEOREM. A square matrix A has a inverse if and only if
      one of the following holds:
         1. rref(A) = I
         2. Ax=0 has unique solution x=0.
         3. det(A) is not zero.
         4. rank(A) = n =row dimension of A.
         5. There are no free variables in the last frame.
         6. All variables in the last frame are lead variables.
         7. nullity(A)=0.

      THEOREM. The inverse matrix is unique and written A^(-1).
      THEOREM. If A, B are square and AB = I, then BA = I.
      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).
      THEOREM.  The inverse of a 2x2 matrix is given by the formula

                   [a  b]        1    [ d  -b]
           inverse [    ]  =  ------- [      ]
                   [c  d]     ad - bc [-c   a]

      THEOREM.  The inverse B of any square matrix A can be
      found from the frame sequence method

                        augment(A,I)
                         toolkit
                          steps combo, swap, mult
                          .
                          .
                          .
                        augment(I,B)

     in which the inverse B of A is read-off from the right panel of
     the last frame.


Slides: Inverse matrix, frame sequence method (71.6 K, pdf, 02 Oct 2009)
Slides: Matrix add, scalar multiply and matrix multiply (122.5 K, pdf, 02 Oct 2009)
Elementary matrices 
   Inverses of elementary matrices.
   Solving B=E3 E2 E1 A for matrix A = (E3 E2 E1)^(-1) B.
   About problem 3.5-44
      This problem is the basis for the fundamental theorem on
      elementary matrices (see below). While 3.5-44 is a difficult
      technical proof, the extra credit problems on this subject
      replace the proofs by a calculation. See Xc3.5-44a and Xc3.5-44b.
   How to do 3.5-16 in maple.
     with(linalg):#3.5-16
     A:=matrix([[1,-3,-3],[-1,1,2],[2,-3,-3]]);
     B:=inverse(A); # expected answer
     A1:=augment(A,diag(1,1,1));
     rref(A1); # Expected answer in right panel
     A2:=addrow(A1,1,2,1);
     A3:=addrow(A2,1,3,-2);
 evalm(A&*B);
Lecture
   Ideas of rank, nullity, dimension in examples.

Slides: Rank, nullity and elimination (111.6 K, pdf, 29 Sep 2009) More on Rank, Nullity dimension 3 possibilities elimination algorithm
Question answered: What did I just do, by finding rref(A)? Problems 3.4-17 to 3.4-22 are homogeneous systems Ax=0 with A in reduced echelon form. Apply the last frame algorithm then write the general solution in vector form.

10-11 Feb: Murphy and Cox

Exam 1 review, questions and examples on exam problems 1,2,3,4,5.
Murphy's Lecture, time permitting: Maple Lab 2, problems 1,2,3 details.
The main lecture had a maple demo of the maple integration in problem 1, along with the introduction to the subject of the lab. This covered Newton Cooling, the hot tea example, and derivation of the DE used in maple lab 2.
Exam 1 date is Feb 17, 2-4pm in WEB 104 or Feb 18, 6:50am in JTB 140.
Sample Exam: Exam 1 key from F2009. See also S2009, exam 1.
Answer Key: Exam 1, f2009, 7:30am (62.9 K, pdf, 30 Sep 2009)
Answer Key: Exam 1, f2009, 12:25pm (339.3 K, pdf, 11 Oct 2009)
Answer Key: Exam 1, S2009, 7:30am (395.7 K, pdf, 02 Mar 2009)
Answer Key: Exam 1, S2009, 10:45am (310.2 K, pdf, 02 Mar 2009)

11 Feb: Determinants. Section 3.6.

See problem notes chapter 3
html: Problem notes F2009 (0.0 K, html, 31 Dec 1969)Lecture: Section 3.6, determinant theory and Cramer's rule.
    slides for 3.6 determinant theory
    Slides: Determinants 2008 (188.3 K, pdf, 26 Apr 2010)
    Manuscript: Determinants, Cramer's rule, Cayley-Hamilton (186.5 K, pdf, 09 Aug 2009)
    Lecture: Methods for computing a determinant.
  1. Sarrus' rule, 2x2 and 3x3 cases.
  2. Four rules for determinants
    1. Triangular Rule (one-arrow Sarrus' Rule): The determinant of a triangular matrix is the product of the diagonal elements.
    2. Multiply rule: B=answer after mult(t,m), then |A| = (1/m) |B|
    3. Swap rule: B=answer after swap(s,t), then |A| = (-1) |B|
    4. Combo rule: B=answer after combo(s,t,c), then |A| = |B|
  3. Cofactor expansion. Details for the 3x3 case.
  4. Hybrid methods.
THEOREM. The 4 rules for computing any determinant can be compressed into two rules,
  1. The triangular rule, and
  2. det(EA)=det(E)det(A)
where E is an elementary combo, swap or mult matrix. Examples: Triangular rule [one-arrow Sarrus' rule], combo, swap and mult rules. Example: Cofactor rule.
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).
<
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 next time.
 Hybrid methods to evaluate det(A) next time.
 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
 
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.

Slides: Determinants 2008 (188.3 K, pdf, 26 Apr 2010)
Manuscript: Determinants, Cramer's rule, Cayley-Hamilton (186.5 K, pdf, 09 Aug 2009)
html: Problem notes F2009 (0.0 K, html, 31 Dec 1969)

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 Page 149+, Exercises 3.1 to 3.6 (869.6 K, pdf, 25 Sep 2003)
    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 (100.3 K, pdf, 23 Sep 2009)
    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 (94.2 K, pdf, 01 Jan 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 S2010 (4.4 K, html, 31 Jan 2010)
    Slides: Determinants 2008 (188.3 K, pdf, 26 Apr 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 (87.2 K, pdf, 10 Mar 2008)
    Slides: Partial fraction theory (121.5 K, pdf, 30 Aug 2009)
    Slides: The pivot theorem and applications (131.9 K, pdf, 02 Oct 2009)
    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)