Linear Algebra
Fall Semester 1998

  • Prerequisites
  • Course Outline
  • Grading
  • Tentative Daily Schedule
  • Text: Linear Algebra with Applications,by Bernard Kolman

    Time: 9:40-10:30 NS 205

    Instructor: Prof. Nick Korevaar

    Web Page:http://www.math.

    Office: JWB 218

    Telephone: 581-7318

    Office hours: M 2-2:50 p.m., T 12:30-2:30 p.m., W 2-2:50 p.m., F 8:20-9:00 a.m.


    Math 1210-1220, first year Calculus. (This was Math 111-112-113 last year.) Previous exposure to vectors, either in a multivariable Calculus course or in a Physics course, is helpful but not essential.


    Course outline:

    This is the first semester in a year-long sequence devoted to linear mathematics. Our topic this semester is linear algebra, a fundamental area of mathematics which is used to describe and study a multitude of subjects in science and life. The origins of this field go back to the algebra which one must solve to find the intersection of two lines in a plane, or of several planes in space, or more generally the solution set of one or more simultaneous ``linear'' equations involving several variables.

    The course begins by studying linear systems of equations and the Gauss-Jordan method of systematically solving them. We see how to write these problems more succinctly in matrix form, the algebra of matrix operations, about inverses of non-singular square matrices, about determinants and their usefulness in solving linear problems. These topics comprise chapters 1-2 of the text.

    In chapter 3 we review the linear geometry of R2,R3, and Rn and discuss the geometric meaning of matrices and determinants, as well as the dot and cross products and their geometric meanings. After chapter 3 we skip to the application called linear programming (chapter 7), which is used heavily in business.

    At this point the course takes a turn towards the abstract as we study general vector spaces in chapters 4 and 6. Basically a vector space is a collection of objects (called vectors) which you can add and scalar multiply, such that certain arithmetric properties hold. From these arithmetric properties one develops notions such as linear independence, bases, dimension, subspaces, coordinates with respect to a basis, change of basis, linear transformations between vector spaces, kernel and range subspaces of linear transformations. We usually visualize vectors in the concrete example of Rn, but in fact there are very natural spaces of functions and of solutions to certain (homogeneous linear) differential equations which are also vector spaces, so that these abstract concepts also apply to them. It is precisely because vector spaces appear in these different disguises that it is worthwhile to discuss them in this abstract way: as characterized by properties rather than by explicit descriptions. You will appreciate this more when you take Math 2280 and apply vector space theory to your study of linear differential equations.

    We will discuss the notion of eigenvectors and eigenvalues for linear transformations from Rn to Rn,in chapter. These will also be used heavily in Math 2280.

    The dot product in Rn lets one talk about orthogonality and orthogonal projections, and we discuss several applications related to this circle of ideas: Gram-Schmidt orthogonalization, methods of least squares (8.4), diagonalizing quadratic forms (8.8), rotating space to express conic sections and quadratic surfaces optimally (8.9-8.10).

    We will show there is also a natural ``dot'' product on certain function spaces, and use this to motivate Fourier series (appendix B.1) which you will return to in Math 2280.

    Although we will use mostly real (scalar) vector spaces, it is often important in applications to allow complex numbers as your scalars, and we discuss this in Appendix A. If there is time at the end of the semester we will also spend a couple of lectures sketching some of the applications of our work to differential equations (8.6), as a way of forshadowing your work in Math 2280.

    There will be approximately 6 computer projects during the semester, to enhance and expand upon the material in the text. They will be written in the software package MAPLE. On MAPLE days we will meet in the Math Department Computer Lab located in Building 129, between JWB and LCB. We do not assume you have had any previous experience with this software and we will make the necessary introductions during the first visit to the lab.



    There will be two midterms, a comprehensive final examination, and homework. Each midterm will count for 20% of your grade, homework will count for 30%, and the final exam will make up the remaining 30%. The book homework will be assigned daily and collected weekly, on Fridays. Maple projects will generally be due one week after they are assigned. A homework grader will partially grade your assignments. The value of carefully working homework problems is that mathematics (like anything) must be practiced and experienced to be learned. It is the Math Department policy, and mine as well, to grant any withdrawl request until the University deadline of October 23.


    ADA Statement:

    The American with Disabilities Act requires that reasonable accomodations be provided for students with physical, sensory, cognitive, systemic, learning, and psychiatric disabilities. Please contact me at the beginning of the semester to discuss any such accommodations for the course.


    Tentative Daily Schedule

    exam dates fixed, daily subject matter approximated

    F28 Aug1.1linear systems
    M31 Aug1.2matrices
    T1 Sept1.3matrix multiplication
    W2 Sept1.4matrix operations
    F4 Sept1.5solving linear systems
    M7 SeptnoneLabor Day
    T8 Septmapleproject I
    W9 Sept1.6matrix inverses
    F11 Sept2.1determinant definition
    M14 Sept2.2cofactor expansions and applications
    T15 Sept2.2" "
    W16 Sept3.1vectors in the plane
    F18 Sept3.2n-vectors
    M21 Sept3.3-3.4linear transformations
    T22 Septmapleproject II
    W23 Sept3.3-3.4linear transformations
    F25 Sept3.5cross product
    M28 Sept3.6lines and planes
    T29 Sept7.1linear programming problem
    W30 Sept7.2simplex method
    F2 Oct7.2" "
    M5 Octreviewreview
    T6 Octexam 11-3, 7.1-7.2
    W7 Oct4.1real vector spaces
    F9 Octnonefall break day
    M12 Oct4.2subspaces
    T13 Oct4.3Linear Independence
    W14 Oct4.4basis and dimension
    F16 Oct4.4basis and dimension
    M19 Oct4.5homogeneous systems
    T20 Octmapleproject III
    W21 Oct4.6matrix rank
    F23 Oct4.7coordinates, change of basis
    M26 Oct4.8orthonormal bases
    T27 Oct4.8-4.9orthogonal complements
    W28 Oct4.9orthogonal complements
    F30 Oct8.4least squares
    M2 Nov8.4 & B.1inner product spaces IV
    T3 Novmapleproject IV
    W4 NovB.1Fourier series
    F6 NovA.1Complex numbers
    M9 NovA.2Complex linear algebra
    T10 Nov6.1linear transformations
    W11 Nov6.2kernel and range
    F13 Nov6.3matrix of linear transformation
    M16 Nov6.3continued
    T17 Novreviewreview
    W18 Novexam 24, 8.4,A,B.1,6
    F20 Nov5.1eigenvalues and eigenvectors
    M23 Nov5.2diagonalization continued
    T24 Novmapleproject V
    W25 Nov5.2diagonalization continued
    F27 NovnoneThanksgiving
    M30 Nov8.8quadratic forms
    T1 Decmapleproject VI
    W2 Dec8.9-8.10conic sections
    F4 Dec8.9-8.10quadric surfaces
    M7 Dec8.10quadric surfaces
    T8 Dec8.6differential equations
    W9 Dec8.6differential equations
    F11 Decallentire course
    T15 DecFINAL EXAMentire course 10:00-12:00