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Lecture and Assignment Schedule
Spring 2004
MATH 308
Lecture Topics Problems Due
Date		 Optional extras
Mar 29
    JRA 1.1-2
  • Systems of Linear Equations
    • Solve by Gauss (forward) elimination of variables moving from left to right, top to bottom and back elimination, from right to left, bottom to top.
    • Represent  by (augmented) matrices
  • Equivalence of systems and row equivalence of matrices:
    • Changes to equations or rows  which don't change the solutions of the associated system
    • Swap, scale, add multiple of one to another.
  • Echelon Form  matrices: What the associated matrix looks like after elimination
  • Read the supplement and errata for the text before working the problems.
  • Complete the web based student information form before 9AM Fri Apr. 2
  • 1.1,  1(Why? Refer to (1),p.2), 5(Why?), 11, 12,  23, 27, 35, 38
  • 1.2,    37, 39, 
    		•  Mar. 31 
  • 1.1, 13,  19, 24,  30, 31, 33, 39 
  • 1.2, 36, 39
  • Read Appendix A1-5 and start using Matlab. 

    •  
    Mar. 31
      More on JRA 1.2
    • Reduced Echelon Form(RREF) and Gauss Elimination
    • Solving a system whose associated matrix is in RREF 
    • Systems with no solutions: inconsistent systems of linear equations.
    • General solution to linear system involves unconstrained and constrained variables.
  • 1.2,  1,  5, 15, 21, 23, 31, 33, 41, 50, 53,  56(In 15, 21  "explain why"  not  "state that", i.e. give a solution not just an answer )
    		•  Apr. 2
  • 1.2, 11, 12, Transform 18 into RREF, 29, 43, 45,  51,  54
  • A2.1,A3.1,A5.1,2

    • Note a Matlab "diary" print out of A3.1 would constitute a solution to 1.2, 30, 31
    Apr. 2
    • Consistent Systems of Linear Equations.
      • Structure of (RR)EF for augmented matrix of system with solutions
      • Fewer non-zero rows r in (RR)EF than variables n in equations
      • Fewer equations than unknowns means either inconsistent or infinitely many solutions
      • Homogeneous equations, RHS all 0, are consistent
  • 1.3, (In the question for Exercises 1-4 replace "independent" by "unconstrained" in both occurances) 1,  3,  5,  8,  9,  12, 13,  15,  19,  21(Why?),  23(Why?),  25, 32
    		•  Apr. 5
  • 1.3, 7,  11, 16, 27,  28, 
    • Lecture Topics Problems Due
    Date    		 Optional extras
    Apr. 5
      JRA 1.4 
    • NetworksApplications:
      • Electrical Networks
      • Traffic flow
      • Quiz 1 on 1.1-1.2
  • 1.4,  1(See errata for error in book answer),  6,  9, 
    		• Apr. 7
  • 1.4, 2,  5,  10 
    		•
    Apr. 7  JRA 1.5
    • Matrix Operations
      • Sums and scalar multiples
      • Vector form of general solution
      • Rn : A vector space
      • Matrix times vector Ax, just like coefficients times variables in equations
      • Matrix times matrix AB= A[B1...Bn] = [AB1....ABn] where Bi are columns of B
  • 1.5,  1(c,d),  5, 13,  17,  43, 44, 45, 47, 49 (In 13,17  "show"  not  "state")
  • 1.5,  9, 21, 31,  52,  55,  56,   57(Also decide which of the two calculations - P(Px) and (PP)x requires more work/multiplications), 60,  63
    		•  Apr. 9
  • 1.5, 35,   53,  58,59, 61, 67, 66, 68, 70, 71
    		•
    Apr. 9 JRA 1.6
    • Properties of Matrix Operations
      • Grouping and order of terms do not matter for matrix addition
      • Grouping does not matter for matrix multiplication
      • Order matters for matrix multiplication
      • Transposes and symmetry
      • Powers
      • (AB)T = BTAT
      • Identity I=[e1...en]
      • Can you cancel: When does AB=AC yield B=C? (Think about unique solutions!)
      • Size (norm) of vectors
  • 1.6,  3,  7,  15,  27,  31, 33, 35, 48, 57
    		•
  • Apr. 12
    		•
  • 1.6, 1,  11,  13,  21, 24,28, 30, 32, 41, 50, 60, 62(b)
    		•
    Lecture Topics Problems Due
    Date    		 Optional extras
    Apr. 12 JRA 1.7. 
    • Linear Independence and Non-singular matrices
      • Linear combinations
      • Zero vectors
      • Linear independence
      • Non-singular square matrices
      • Unit vectors
      • Recognizing dependent sets
      • p vectors in m space, p>m, are dependent
  • 1.7,  1,  2,  9,  17,  24,   27,  35, 47,  50,
    		•  Apr. 16
  • 1.7,   6,  9,  18, 22,  25,  53, 55,  58
    		•
    Apr. 14 JRA 1.8
      Applications:
      • Data fitting
      • Numerical Integration
  • 1.8, 6,  12, 19 
    		•  Apr. 16
  • 1.8,  1, 7, 8, 9, 10, 11, 13, 25, 27 31, 34
    		•
    Apr. 16 JRA 1.9
    Matrix inverses
    • Non-singular matrices and unique solutions. Theorem 13 in 1.7
    • Definition
    • Calculating inverses
    • Uses of inverses
    • Existence of inverses
    • Properties (Thm. 17) including (AB)-1=B-1A-1
    • (While important, omit Ill-conditioned matrices)
  • 1.9,  3,  7, 11, 19,   22, 38, 41, 54, 58, 68,  70,  73(Add the word "singular" just before "matrix" in 73)
    		•  Apr. 23
  • 1.9, 1,  6,  17, 25, 27, 33, 50, 52, 67,  72, 
  • Supplemental and conceptual Exercises. Matlab exercise #1, p. 108 extends 1.5.57.
    		•
    Lecture Topics Problems Due
    Date    		 Optional extras
    Apr. 19 Quiz on 1.3-1.8
    JRA 3.1 and 3.2
  • Geometric vectors in R2 and R3  and their algebraic properties: 
      • Addition
      • Scalar multiplication
    • Subsets defined by 
      • Geometric properties
      • Linear or non-linear equations
    Intro to 3.2 - Algebraic properties of n-tuples of numbers    		 3.1,  5,  7,  8,  19,  23,  25,  27 Apr. 23  
    Apr. 21 JRA 3.2 
    • Vector space properties of Rn
      • Zero vector
      • Sums
      • Scalar multiples
      • Order of summing doen't matter
      • Grouping of summands doesn't matter
    • Subspaces: 

      • Subsets which contain the zero vector and all sums and scalar multiples of vectors in the set.
    		 3.2,  1,  7,  9,  11, 15,   18,  19,  28,  30,  32 ("union", U ∪ V, means "all in U or V") Apr. 23  
    Apr. 23 JRA 3.3 - Examples of Subspaces
    • Span of a subset - all linear combinations of vectors in subset. Smallest subspace containing subset
    • Null space of a matrix A - all  x which solve homogeneous equation Ax=0
    • Range of a matrix A: all y for which the equation  Ax=y has some solution.
    		 3.3,  15,  19,  21(a,b,c),  25,  35,  40 Apr 30  
    Lecture Topics Problems Due
    Date    		 Optional extras
    Apr. 26 More on JRA 3.3, Subspace examples.
    • Column space of a matrix A - Sp({columns of A}) = R(A)
    • Row space  of a matrix A -- the span of the rows of A.
      • Row equivalent matrices A and B have the same row space.
    		 3.3,  33,  37,  47,  50,  51, 52,  53 Apr 30  
    Apr 28 JRA   3.4 Bases for Subspaces
    • Spanning set for a subspace W:  Sp(S)=W
    • Minimal spanning set: leave out any vector and it no longer spans
    • Basis for W is a linearly independent spanning set.
    • Standard basis for Rn: e1, ..., en , ei has 1 in i'th row.
    • Coordinates  with respect to a basis
    		 3.4,  1,  9(b,c),  11,  23(a),  , 28, 31(Alternate hint: Use Theorem 18, 1.9),  36, 37 Apr 30  3.4,  7, 9(a,d),   38,  39
    Apr 30 JRA 3.5 Dimension:  Number of vectors in a basis.
    • Any  set of p+1 vectors in a subspace spanned by p vectors is linearly dependent
    • Any two bases for the same space have the same number of vectors.
    • rank(A)=dim(R(A)), nullity(A) = dim(N(A))
    • rank(A)+nullity(A)=#columns of A
    		 3.5,  5,  9,  13,  17,  23,  27(a),  31  34,  35,  38, May 7 3.5,  1,  3,   7,  18,27(b),  29
    Lecture Topics Problems Due
    Date    		 Optional extras
    May 3 Quiz on 1.9,3.1-3.4
    Continuation of 3.5    		 Previous assignment    
    May 5 JRA 3.6 Orthogonal bases.
    • Define orthogonal (perpendicular) xTy = 0
    • Set of mutually orthogonal  non-zero vectors is independent.
    • Orthogonal bases:
      • Finding coordinates is easy - take dot product, i.e. to solve b=x1u1 + ... +xp up just use orthogonality to get xi = uiTb/uiTui
      • Construction by orthogonalization (based on  orthogonal  projection)
    		 3.6,  2,  5,  10,  13,  20,  21,  22, 23, 28  May 7 3.6, 1, 8, 9, 14, 15,   24-26
    May 7 JRA 3.7: Linear Transformations
    • Function T:V-->W with 
      • T(u + v) = T(u) + T(v)
      • T(au) = a T(u)
    • Main example: Multiplication by a matrix A.  TA(x) = Ax
    • Geometric examples: Orthogonal projection on a subspace W with an orthogonal basis {u1,u2,...up}, T(b)=x1u1+...+xpup with  xi = uiTb/uiTui .

      • Also: Rotations and reflections.
    • Matrix of a lin. trasformation T:Rn-->Rm, [T]=[T(e1)...T(en)], i.e. column i of [T] is T(ei)
    		 3.7,  2,  6,  7,  8,  10,  18,  19, 20(Hint:First find the matrix A of T by solving the equation A[v1v2]=[u1u2]), 29, 34, 45(b), 46(b)  May 14 3.7,  1,  3,  4,5, 11,  12,  15, 17, 23,  25,  33
    Lecture Topics Problems Due
    Date    		 Optional extras
    May 10 JRA 3.8: Least-Squares Solutions to inconsistent systems.
    • Given a matrix A and a vector b, the a least squares solution to Ax = b is any x* for which ||Ax*-b|| < or = ||Ax-b|| for all x.
    • Geometrically: Drop a perpendicular from b to the R(A), then the coefficients of the linear combination of the columns of A at the base of the perpendicular give the least squares solution
    • x* solves the equation ATAx=ATb (or more geometrically, AT(Ax-b)=0) as these equations express that each column of A is perpendicular to Ax-b
    • Examples and applications:

      • low degree polynomial fits to lots of data, linear fits to more than 2 data points, quadratic fits to more than 3 data points
    		 3.8,  1,  3,  7,  9, 11,  12 May 14  3.8, 17, 
    May 12 JRA 3.9 Theory and pratice of least squares.
    How and why the middle two bullets from May 10 work.    		 3.9,  1,  3, 8,  11, 16,  Do but not to hand in. Solutions available May 14 Ch. 3 Supplement. and Conceptual exercises
    3.9, Matlab Exercises, p.270, 1
    May 14 The Eigenvalue Problem - JRA 4.1
  • Eigenvalue for A: scalar λ with Ax= λx for some vector x≠θ
  • Eigenvector for A: x≠θ such that Ax= λx i.e. Directions x such that multiplication by A is just a "stretching" (multiplication) by a scalar λ.
    		•  4.1,  1,  3,  7,  15,  17,  19 May 21  4.1,  18,
    Lecture Topics Problems Due
    Date    		 Optional extras
    May 17 Quiz on 3.4-3.9
    JRA 4.2    		 4.2,  2,  7,  9,  17,  24,  25,  27, 29,  34 May 21 4.2,  11,  15, 26, 28, 30
    May 19 4.3
    4.3, 3, 7,  9,  13,  16,  19,  23, 26    		 May 21 4.3,  14,  17,  25
    May 21 4.4 4.4,  5,  7,  11,  15,  18,  24,  27 May 26 4.4, 26,  28,  29 
    Hint: A characteristic polynomial ± t n + an-1 tn-1 + ... + a0 with integer coefficients can only have an integer root p when p is an integer divisor of a0, the constant term. Use this to find roots when the degree is 3 or 4 by examining the factors of a0 and using long division or synthetic division to check if t±p divides the polynomial.
    Lecture Topics Problems Due
    Date    		 Optional extras
    May 24 4.5 4.5,  1,  4,  11,  13,  14,  23 , 24,  28,  29 Do but do not hand in. Solutions available May 26  
    May 26 4.6 4.6,  21,  23,  25,  29,  37,  39,  Do, but do not hand in. Solutions available May 27  4.6,  1,  5,  7,  13,  17  
    May 28  4.7
  • Project part I(Preliminary) due
  • Quiz on 4.1-4.6
    		• 4.7,  1,  3,  4,  10,  14,  15, 25,  26,  27 Do not hand in. Sol on Jun 1 4.7,  38-41
    Lecture Topics Problems Due
    Date    		 Optional extras
    May 31 -- HOLIDAY
    June 2 4.8 4.8,  1,  3,  7,  9,  15 Sol on June 3 4.8,  23-26
    June 4 Review 
  • Final quiz on 4.6-7
  • Final project due
    		• Final Project -Part I and II  Part II may be postponed until the final   
    June 7-11 is exam week.  Final EXAM covering Chapters 1,3&4 will be held in our classroom on the DATE and TIME listed in the Official UW Exam Schedule

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