## MATHMATICS 2270-1 Linear Algebra Fall Semester 1998

• Prerequisites
• Course Outline
• Tentative Daily Schedule

#### Prerequisites:

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.

TOP

#### 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.

TOP

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.

TOP

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.

TOP

## 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