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2250-1 7:30am Lecture Record Week 6 S2012

Last Modified: February 20, 2012, 21:36 MST.    Today: October 19, 2017, 21:32 MDT.

13 Feb: Elementary matrices. Inverses. Sections 3.4, 3.5.

Inverse matrix
 Definition: A has an inverse B if and only if AB=BA=I.
 THEOREM. An inverse is unique.
 THEOREM. If A has an inverse, then A is square.
          Non-square matrices don't have an inverse.
 THEOREM. The zero matrix does not have an inverse.
 THEOREM. Every elementary matrix E has an inverse. It is found
          as follows:
              Elementary Matrix     Inverse Matrix
              combo(s,t,c)          combo(s,t,-c)
              mult(t,m)             mult(t,1/m)
              swap(s,t)             swap(s,t)
How to compute the inverse matrix 
      Def: AB=BA=I means B is the inverse of A.
      Inverse = adjugate/determinant (2x2 case). See the theorem below for 2x2.
      Inverse from the fundamental theorem on frame sequences.
      Frame sequences method. See the theorem below.
      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
           1      [ d  -b]
       -------    [      ]
        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 (97.9 K, pdf, 03 Mar 2012)
Slides: Matrix add, scalar multiply and matrix multiply (156.7 K, pdf, 03 Mar 2012)

Upcoming on 16 Feb:


Exam 1. Problems 1,2,3.
Sample Exam: Exam 1 keys from S2010 and F2010.
HTML: Exam links for the past 5 years (18.9 K, html, 02 May 2012)
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 (227.1 K, pdf, 03 Mar 2012)
    Manuscript: Determinants, Cramer's rule, Cayley-Hamilton (326.0 K, pdf, 07 Feb 2012)

14 Feb: Determinants, Survey of Results. Section 3.6

  How to do 3.5-16 in maple.
     macro(combo=linalg[addrow]);
     A:=Matrix([[1,-3,-3],[-1,1,2],[2,-3,-3]]);
     B:=A^(-1); # expected answer
     Id:=Matrix([[1,0,0],[0,1,0],[0,0,1]]);
     A1:= < A | Id >;
     linalg[rref](A1); # Expected answer in right panel
     A2:=combo(A1,1,2,1);
     A3:=combo(A2,1,3,-2);
     See problem notes chapter 3            

html: Problem notes S2012 (5.2 K, html, 08 Apr 2012)
 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.
 General structure of linear systems.
    Superposition.
    General solution
       X=X0+t1 X1 + t2 X2 + ... + tn Xn.
   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.
 
   EXAMPLE. A 3x3 matrix, frame sequence, Special Solutions of Gilbert Strang.
            How to find the vector general solution.
            How to find x_p: Set all free variable symbols to zero
            How to find x_h: Take all linear combinations of the 
                             Special Solutions.
            Superposition: x = x_p + x_h = General Solution
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|

15 Feb: Cramers Rule, Adjugate formula. Section 3.6

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). 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).
Discussion of 3.5 problems.
  Lecture
   Ideas of rank, nullity, dimension in examples.

Slides: Rank, nullity and elimination (154.1 K, pdf, 03 Mar 2012) More on Rank, Nullity dimension 3 possibilities elimination algorithm
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 (227.1 K, pdf, 03 Mar 2012)
Manuscript: Determinants, Cramer's rule, Cayley-Hamilton (326.0 K, pdf, 07 Feb 2012)
html: Problem notes S2012 (5.2 K, html, 08 Apr 2012)

16 Feb: Exam 1

Exam 1. Problems 1,2,3.
Sample Exam: Exam 1 keys from S2010 and F2010.
HTML: Exam links for the past 5 years (18.9 K, html, 02 May 2012)

17 Feb: Problem session 3.4, 3.5, 3.6

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
 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 (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 2010 (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)