MATH 221-02

Linear Algebra

  • Class Meets: 1:10-2:00 MWF in Wilson 1-132

  • Instructor:
    Curt Vogel
    2-210 Wilson Hall
    ph. 994-5332
    E-mail: vogel@math.montana.edu
    Home Page: www.math.montana.edu/~vogel

  • Textbook:
    Matrix Theory and Modelling, by Ken Bowers and Gary Bogar (required and available at Cards & Copies)
    Handouts for M221 prepared by Todd Shaw

  • Software:
    MATLAB
    Web-based MATLAB tutorial
    Tutorial on UNIX operating system (for the workstations in the computer lab)

  • Course Outline:
    See Table of Contents of textbook.

  • Grading:
    2 in-class exams, to be given after Chapter 1 and Chapter 2 are covered (2 X 25%); Quizes, given roughly once a week (20%); Final exam (30%). Recommended homework will be assigned, but not collected or graded.

    The final exam is scheduled for 8-9:50 am, Tues, May 6, 2003.

    The final will be given in the usual classroom (1-132 Wilson)

    Students are strongly encouraged to do the recommended homework and to work together.

  • Quiz 1 sheduled for Friday, 1/24/03:
    May cover scalar multiplication, vector addition, vector magnitude (length), dot product, and linear combinations.

  • Recommended Homework, 1/24/03:
    Exercises 1.2.1, 1.2.3, 1.2.8, 1.2.13, 1.2.17, 1.2.21, 1.2.27

  • Recommended Homework, 1/27/03 & 1/29/03:
    Exercises 1.3.1, 1.3.5, 1.3.11, 1.3.12, 1.3.18, 1.3.19, 1.3.21, 1.3.22, 1.3.23, 1.3.24, 1.3.31, 1.3.34, 1.3.47, 1.3.48
    Solve the linear system 2x - z = 3, 2y - 4z = 3, -2z = 8. (It is upper triangular, so use back substitution). Also, write the system in matrix-vector form, and observe the coefficient matrix is upper triangular.

  • Quiz 2 sheduled for Friday, 1/31/03:
    May cover matrix operations (addition, multiplication, transpose) and back substitution.

  • Recommended Homework, 2/3/03:
    Exercises 1.4.1, 1.4.2, 1.4.3, 1.4.6, 1.4.11, 1.4.13, 1.4.15, 1.4.16, 1.4.18

  • Recommended Homework, 2/5/03:
    In each of exercies 1.5.1, 1.5.2, 1.5.3, 1.5.5, 1.5.7, 1.5.9, 1.5.11, do the following:
    (i) Convert to reduced row-echelon form, if it is not already in that form.
    (ii) Either state that there is no solution; find the unique solution; or find the general solution in the form
    x = particular soln + homogeneous soln.

  • Quiz 3 sheduled for Friday, 2/7/03:
    May cover
    (i) conversion to row-echelon form or reduced row-echelon form using elementary row operations.
    (ii) determining if their there is no solution, or finding the unique solution, or finding the general solution to Ax=b.

  • Quiz 4 sheduled for Friday, 2/14/03:
    May cover
    (i) Computation of determinants.
    (ii) Computation of matrix inverses.

  • Recommended Homework, 2/19/03, 2/21/03:
    Exercise 1.8.2, 1.8.6, 1.8.10.

  • Quiz 5 sheduled for Monday, 2/24/03:
    May cover
    (i) Permutation matrices.
    (ii) Computation of P,L,U factors of matrix A.
    (iii) Solution to Ax=b, given the P,L,U factors.

  • Exam I sheduled for Friday, 2/28/03:
    Exam topics will be announced later.

  • Recommended Homework, 3/3/03, 3/5/03:
    Section 2.1, Exercises 1abcdeg, 4, 5, 6ab, 10

  • Quiz 6 sheduled for Friday, 3/7/03:
    May cover subspaces and span.

  • Recommended Homework, 3/17/03, 3/19/03:
    Section 2.2, Exercises 1a, 3abcd (ignore part about "what subspace"), 4ab, 6, 10, 11 (ignore part about "extend basis ..."), 12abc.

  • Quiz 7 sheduled for Friday, 3/21/03:
    May cover linear independence and basis.

  • Recommended Homework, 3/24/03, 3/26/03:
    Section 2.3, Exercises 1,2,3abc, 4, 8abdf.
    More abstract "challenge" problem in Section 2.3: Exercise 5.

  • Quiz 8 sheduled for Friday, 3/28/03:
    May cover rank; null, row, and column spaces; and related material.

  • Recommended Homework, 4/2/03, 4/4/03:
    Section 2.4, Exercises 1acd,2abc,3,4,,7ab,10ac,11ab,14ab,18.
    Section 2.6, Exercises 4a, 6a, 9.

  • Quiz 9 sheduled for Monday, 4/7/03:
    May cover orthogonality, (vector and matrix) projection, least squares fit to data.

  • Exam II sheduled for Friday, 4/11/03:
    May cover subspace; span; linear independence; basis; dimension; rank; null, row, and column spaces; orthogonality, (vector and matrix) projection, least squares fit to data.

  • Recommended Homework, 4/14/03, 4/16/03:
    Section 3.1, Exercises 2abcg,4ac,6ac,8,9

  • Quiz 10 sheduled for Monday, 4/21/03:
    May cover eigenvalue and eigenvector computations.

  • Recommended Homework, 4/21/03:
    1. Let A = [2/3 1/4; 1/3 3/4] (A is 2X2. Semicolon (;) separates rows)
    Compute the limit as n -> infinity of A^n v (nth power of A times
    vector v) for almost any vector v.
    2. Use the eigencomposition and then the matrix exponential to
    solve the following ODE initial value problem:
    x'(t) = x(t) + 4y(t), t>0
    y'(t) = 2x(t) - y(t)
    x(0) = 1, y(0) = 0.
    See class handout of 4/23/03 for solutions.

  • Recommended Homework, 4/25/03:
    If lambda,v are an eigenvalue, eigenvector pair for a matrix A and A is nonsingular, show that (i) lambda cannot be zero; and (ii) 1/lambda, v are an eigenvalue, eigenvector pair for the inverse of A.
    If A is (upper or lower) triangular, show that the eigenvalues of A are the diagonal entries of A.
    Section 3.3, Exercises 3,5cd,6be,7. Anywhere that you read "find a matrix P", replace this by "find an eigendecomposition".

  • Quiz 11 sheduled for Monday, 4/28/03:
    May cover applications of the eigendecomposition like computing
    limit as n -> infinity of A^n x and solving ODE IVP's.
    May also cover geometric multiplicity and (non)degenerate matrices.