Time: 9:40-10:30 NS 205
Instructor: Prof. Nick Korevaar
Web Page:http://www.math. utah.edu/~korevaar
Office: JWB 218
Telephone: 581-7318
Email:korevaar@math.utah.edu
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.
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.
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.
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.
Grading:
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.
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Tentative Daily Schedule
exam dates fixed, daily subject
matter approximated
F 28 Aug 1.1 linear systems M 31 Aug 1.2 matrices T 1 Sept 1.3 matrix multiplication W 2 Sept 1.4 matrix operations F 4 Sept 1.5 solving linear systems M 7 Sept none Labor Day T 8 Sept maple project I W 9 Sept 1.6 matrix inverses F 11 Sept 2.1 determinant definition M 14 Sept 2.2 cofactor expansions and
applications T 15 Sept 2.2 " " W 16 Sept 3.1 vectors in the plane F 18 Sept 3.2 n-vectors M 21 Sept 3.3-3.4 linear transformations T 22 Sept maple project II W 23 Sept 3.3-3.4 linear transformations F 25 Sept 3.5 cross product M 28 Sept 3.6 lines and planes T 29 Sept 7.1 linear programming problem W 30 Sept 7.2 simplex method F 2 Oct 7.2 " " M 5 Oct review review T 6 Oct exam 1 1-3, 7.1-7.2 W 7 Oct 4.1 real vector spaces F 9 Oct none fall break day M 12 Oct 4.2 subspaces T 13 Oct 4.3 Linear Independence W 14 Oct 4.4 basis and dimension F 16 Oct 4.4 basis and dimension M 19 Oct 4.5 homogeneous systems T 20 Oct maple project III W 21 Oct 4.6 matrix rank F 23 Oct 4.7 coordinates, change of basis M 26 Oct 4.8 orthonormal bases T 27 Oct 4.8-4.9 orthogonal complements W 28 Oct 4.9 orthogonal complements F 30 Oct 8.4 least squares M 2 Nov 8.4 & B.1 inner product spaces IV T 3 Nov maple project IV W 4 Nov B.1 Fourier series F 6 Nov A.1 Complex numbers M 9 Nov A.2 Complex linear algebra T 10 Nov 6.1 linear transformations W 11 Nov 6.2 kernel and range F 13 Nov 6.3 matrix of linear transformation M 16 Nov 6.3 continued T 17 Nov review review W 18 Nov exam 2 4, 8.4,A,B.1,6 F 20 Nov 5.1 eigenvalues and eigenvectors M 23 Nov 5.2 diagonalization continued T 24 Nov maple project V W 25 Nov 5.2 diagonalization continued F 27 Nov none Thanksgiving M 30 Nov 8.8 quadratic forms T 1 Dec maple project VI W 2 Dec 8.9-8.10 conic sections F 4 Dec 8.9-8.10 quadric surfaces M 7 Dec 8.10 quadric surfaces T 8 Dec 8.6 differential equations W 9 Dec 8.6 differential equations F 11 Dec all entire course
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T 15 Dec FINAL EXAM entire course 10:00-12:00