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2250 7:30am Lectures Week 6 S2014

Last Modified: February 09, 2014, 20:49 MST.    Today: October 22, 2017, 02:23 MDT.
 Edwards-Penney, sections 4.1, 4.2, 4.3, 4.4
  The textbook topics, definitions and theorems
Edwards-Penney 4.1, 4.2, 4.3, 4.4 (5.6 K, txt, 19 Dec 2013)
Edwards-Penney 4.5, 4.6, 4.7 (7.0 K, txt, 19 Dec 2013)

Monday: Cofactor Expansion, Inverse Formula, Determinant Product Theorem 3.6

 Discussion of 3.5 problems.
  Lecture
   Ideas of rank, nullity, dimension in examples.
Slides: Rank, nullity and elimination (156.3 K, pdf, 21 Dec 2012) Lecture on Cofactor Method 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. Adjugate Matrix. How to form the matrix of cofactors and its transpose. DEF: The adjugate matrix. THEOREM. The 4 rules for computing any determinant can be compressed into two rules, det(triangular matrix)=the product of the diagonal elements, det(EA)=det(E)det(A), where E is an elementary matrix, combo, swap or mult. 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 S2014 (4.9 K, html, 10 Dec 2013) REVIEW: Transpose matrix (A^T)^T = A (A + B)^T = A^T + B^T (AB)^T = B^T A^T det(A)=det(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
New Topics
  Rank, Nullity, Dimension and Elimination for Equations
      Three possibilities
      Definitions: rank, nullity, dimension
      Rank-Nullity theorem
      Elimination algorithm
      Examples
  References:
Slides: Rank, nullity and elimination (156.3 K, pdf, 21 Dec 2012) Ch 3 PROBLEMS Problems 3.3-10,20 using maple Problem 3.4-20 Long details in FAQ 3.4 Problem 3.4-30, Cayley-Hamilton Problem 3.4-29 is used in Problem 3.4-30. See FAQ 3.4 for details Cayley-Hamilton Theorem. It is a famous result in linear algebra which is the basis for solving systems of differential equations. Discussion
Slides: Cayley-Hamilton Theorem (100.5 K, pdf, 21 Dec 2012)
Manuscript: Determinants, Cramer's rule, Cayley-Hamilton (326.0 K, pdf, 07 Feb 2012)

Tuesday: Vector spaces R^3 and R^n. Sections 4.1, 4.2.

Exercises
  Problem 3.4-40, Superposition proof
     The problem is to prove the superposition principle for the
     matrix equation Ax=b. It is the analog of the differential
     equation relation y=y_h + y_p. Details in FAQ 3.4.
   Problems 3.5-16,26,44
       For the 3.5-44 proof, see the 3.5 FAQ.
   Problems 3.6-6,20,32,40,60
     3.6 FAQ for details and answer checks
     Maple Answer Checks: Compute det(A), inverse(A), adjoint(A)
   Review 3.6. matrix A is 10x10 and has 92 ones. What's det(A)?
   Problem 3.6-60, nxn determinants
     (60a) B_n = 2B_{n-1} - B_{n-2}, by cofactor expansion
     (60b) B_n = n+1 by induction
     
Intro to Ch4
  Def: Vector==package of data items
  Vector Toolkit
     The 8-property toolkit for vectors [4.2]
     Reading: Sections 4.1, 4.2 in Edwards-Penney
  Def: vector space, subspace
    Data set == Vector space
    Working set == subspace.
  Examples of vectors:
     Four classical vector models,
       Vectors are not arrows
       Fixed vectors
       Triad i,j,k algebraic calculus model
       Physics and Engineering arrows
       Gibbs vectors.
     Digital photos,
     Fourier coefficients,
     Taylor coefficients,
     Solutions to DE.
       Example: y=2exp(-x^2) for DE y'=-2xy, y(0)=2.
Four Vector Models:
Slides: vector models and vector spaces (144.1 K, pdf, 03 Mar 2012) Parallelogram law. Head minus tail rule. Abstract vector spaces, 4.2. Def: Vector==package of data items. Vectors are not arrows The 8-Property Vector Toolkit Def: abstract vector space Data set == Vector space Def: Subspace of a vector space Working set == smaller vector space = subspace Vector space of color photographs RGB color separation and matrix add Intensity adjustments and scalar multiply Digital photos and matrix add, scalar multiply visualization.
Slides: Digital photos, Maxwell's RGB separations, visualization of matrix add (170.2 K, pdf, 03 Mar 2012)
Slides: More on digital photos, checkerboard analogy (141.9 K, pdf, 03 Mar 2012)

Wednesday: Subspace Tests and Applications. Sections 4.2, 4.3.

  Subspaces
    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.
      Web reference
Slides: Vector space, subspace, independence (168.4 K, pdf, 17 Feb 2014) More on vector spaces and subspaces Detection of subspaces and data sets that are not subspaces. Subspace Theorems: Subspace criterion, Kernel theorem, Not a subspace theorem. The Span Theorem. Preview: Independence-Dependence Theorems Determinant test Rank test Pivot theorem Orthogonal vector theorem Wronskian test for functions Sample test for functions Web references
Slides: Vector space, subspace theorems, independence tests (168.4 K, pdf, 17 Feb 2014)
Slides: Orthogonal vector theorem (124.8 K, pdf, 03 Mar 2012) Use of subspace theorems 1,2 in section 4.2. Subspace 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 (4.2 Theorem 2). Subspace applications. When to use the kernel theorem. When to use the subspace criterion. When to use the not a subspace theorem. Identifying a subspace with the span theorem Identifying a subspace defined by equations Problems 4.1,4.2. Textbook Reading: Chapter 4, sections 4.1 and 4.2. Web references for chapter 4.
Slides: Vector space, subspace, independence (168.4 K, pdf, 17 Feb 2014)
Manuscript: Vector space, Independence, Basis, Dimension, Rank (268.7 K, pdf, 22 Mar 2010)
Slides: The pivot theorem and applications (189.2 K, pdf, 03 Mar 2012)
Slides: Rank, nullity and elimination (156.3 K, pdf, 21 Dec 2012)
Slides: Digital photos, Maxwell's RGB separations, visualization of matrix add (170.2 K, pdf, 03 Mar 2012)
Slides: More on digital photos, checkerboard analogy (141.9 K, pdf, 03 Mar 2012)
Slides: Orthogonality (124.8 K, pdf, 03 Mar 2012)
Transparencies: Ch4 Page 237+ slides, Exercises 4.1 to 4.4, some 4.9 (463.2 K, pdf, 25 Sep 2003)
html: Problem notes S2013 (0.0 K, html, 31 Dec 1969)

Thursday: Rebecca

Exam 1 Review. Problems 1, 2, 3, 4, 5.
 Sample Exam: Exam 1 key from S2012. See also F2010, exam 1.
Sample Midterm 1, S2014, with solutions (1625.5 K, pdf, 11 Mar 2013)
Exams and exam keys for the last 5 years (22.4 K, html, 30 Apr 2014)

Friday: Midterm Exam 1

 Exam 1 starts at 7am, if you need extra time. 
 Otherwise, please arrive at 7:30am, the regular class time.
 The exam is designed for 50 minutes.
 
 The official end of the exam is at 8:20am, but you will be given an extra 5 minutes  for complete whatever is in progress. 
 
 Please vacate the room by 8:30am. The next class begins at 8:35am. 
 
 Kindly exit BEFORE 8:35am, to allow time for setup and lecture preliminaries.
 

References for chapters 3 and 4, Linear Algebra


Manuscript: Linear algebraic equations, no matrices (429.7 K, pdf, 30 Jan 2014)
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 from EP 2nd Edition (869.6 K, pdf, 25 Sep 2003)
Transparencies: Ch4 all, Exercises 4.1 to 4.7, from EP 2nd edition (461.2 K, pdf, 03 Oct 2010)
Transparency: Sample solution ER-1 [exam review and maple lab 3] (184.6 K, jpg, 08 Feb 2008)
Slides: Elementary matrix theorems (161.5 K, pdf, 03 Mar 2012)
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 (237.3 K, pdf, 15 Dec 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 (110.1 K, pdf, 14 Dec 2012)
Slides: Three rules, frame sequence, maple syntax (35.8 K, pdf, 25 Jan 2007)
Jpeg: 3x3 Frame sequence and general solution (315.9 K, jpg, 12 Dec 2012)
Slides: Determinants 2012 (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 (156.3 K, pdf, 21 Dec 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)
Maple: Lab 5, Linear algebra (0.0 K, pdf, 31 Dec 1969)
html: Problem notes S2013 (0.0 K, html, 31 Dec 1969)
Text: Lawrence Page's pagerank algorithm (0.7 K, txt, 06 Oct 2008)
Text: History of telecom companies (1.9 K, txt, 04 Apr 2013)

Monday: Independence and Dependence. Sections 4.1, 4.3, 4.7

Sections 4.1, 4.3 and some part of 4.7.
  Review:
    Is the 8-property vector toolkit good for nothing?
      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. This is a key example
      in the theory of eigenanalysis.
Quick Review of Vector spaces 4.1, 4.2
  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).
   Span Theorem (Theorem 1, 4.3)
   Not a Subspace Theorem (Theorem 1 backwards)
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.
    Three vectors v1, v2, v3.
 Abstract vector space tests
    One vector v1.
    Two vectors v1, v2.
 Algebraic tests.
   Rank test.
   Determinant test.
   Pivot theorem.
   Additional tests
      Sampling test.
      Wronskian test.
      Orthogonal vector test.
 THEOREM: Pivot columns are independent and non-pivot columns
          are linear combinations of the pivot columns.
Web References
Slides: Vector space, subspace, independence tests (168.4 K, pdf, 17 Feb 2014)
Slides: The pivot theorem and applications (189.2 K, pdf, 03 Mar 2012)
Slides: Orthogonality (124.8 K, pdf, 03 Mar 2012)
Manuscript: Vector space, Independence, Basis, Dimension, Rank (268.7 K, pdf, 22 Mar 2010)
Slides: Rank, nullity and elimination (156.3 K, pdf, 21 Dec 2012)
Slides: Base atom, atom, basis for linear DE (117.6 K, pdf, 03 Mar 2012)