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Introductory Course, Syllabus for Linear Algebra | MATH 215, Papers of Linear Algebra

Illinois Wesleyan University (IWU)Linear Algebra

Material Type: Paper; Class: Linear Algebra; Subject: Mathematics; University: Illinois Wesleyan University; Term: Spring 2008;

Typology: Papers

Pre 2010

Uploaded on 08/18/2009

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Linear Algebra, Math 215, Spring 2008
Professor: Lawrence Stout
Office: C209C CNS
Office hours: 11-12 TTh, 1-2 MW 10-11F
Phone: 556-3038
e-mail: lstout@iwu.edu
Web: http://www.iwu.edu/∼lstout
Text: L.N. Stout, Linear Algebra , manuscript 1998,
revised Summer 2007
Course Description Linear Algebra is the study of vector spaces and
linear transformations. This is rather different from your study of calculus or
previous algebra experience because we are studying a whole class of spaces,
not just the properties of one particular number system. Vectors in Euclidean
2-space and 3-space are studied in Calculus 3 and in the analysis sequence,
so some of the properties of a vector space may be familiar: vectors can be
added to get another vector and vectors can be multipied by a scalar to get
another vector. A vector space over a field is a set equipped with a two
operations, one which behaves like addition of vectors, and one that behaves
like multiplication by a scalar. Linear transformations are functions which
respect the operations of addition and scalar multiplication. Differentiation
is a good example of a linear transformation.
Properties of vector spaces and linear transformations are often most eas-
ily proved in the abstract. Once we have the notion of a basis, however, we
can make the finite dimensional part of the subject quite concrete by repre-
senting vectors as n-tuples of numbers and linear transformations as matrices.
Another side of the subject then involves manipulating matrices and inter-
preting the results you get. This linking of the abstract with the concrete
makes the subject a good bridge from the concrete mathematics of the lower
division to the abstract mathematics at the 300 and 400 levels. Concepts
from linear algebra are used in nearly every upper level mathematics course
and have become quite important in physics, engineering, economics, and
statistics.
This course approaches the subject by first limiting the computational
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Download Introductory Course, Syllabus for Linear Algebra | MATH 215 and more Papers Linear Algebra in PDF only on Docsity!

Linear Algebra, Math 215, Spring 2008

Professor: Lawrence Stout Office: C209C CNS Office hours: 11-12 TTh, 1-2 MW 10-11F Phone: 556- e-mail: lstout@iwu.edu Web: http://www.iwu.edu/∼lstout Text: L.N. Stout, Linear Algebra , manuscript 1998, revised Summer 2007

Course Description Linear Algebra is the study of vector spaces and

linear transformations. This is rather different from your study of calculus or previous algebra experience because we are studying a whole class of spaces, not just the properties of one particular number system. Vectors in Euclidean 2-space and 3-space are studied in Calculus 3 and in the analysis sequence, so some of the properties of a vector space may be familiar: vectors can be added to get another vector and vectors can be multipied by a scalar to get another vector. A vector space over a field is a set equipped with a two operations, one which behaves like addition of vectors, and one that behaves like multiplication by a scalar. Linear transformations are functions which respect the operations of addition and scalar multiplication. Differentiation is a good example of a linear transformation. Properties of vector spaces and linear transformations are often most eas- ily proved in the abstract. Once we have the notion of a basis, however, we can make the finite dimensional part of the subject quite concrete by repre- senting vectors as n-tuples of numbers and linear transformations as matrices. Another side of the subject then involves manipulating matrices and inter- preting the results you get. This linking of the abstract with the concrete makes the subject a good bridge from the concrete mathematics of the lower division to the abstract mathematics at the 300 and 400 levels. Concepts from linear algebra are used in nearly every upper level mathematics course and have become quite important in physics, engineering, economics, and statistics. This course approaches the subject by first limiting the computational

difficulty and considering all of the notions of Linear Algebra in the very simple setting of Euclidean 2 space where geometric intuition works nicely and there is little difficulty in solving any systems of equations which may come up. We then move to the level of full abstraction. This necessitates a more systematic approach to solving systems of equations, so we take up Gaussian elimination as a computationally efficient means of systematic solu- tion. The problem of solution of a system of equations can be rather sensitive to round off error, so we will avoid using approximate arithmetic and leave analysis of error propigation and other algorithms which control error to a course in numerical analysis or numerical linear algebra. One particular algorithm gets quite a workout in this course: row re- duction. We will use it first to solve systems of equations, then to answer questions about independence (which reduce to systems of equations, but we’ll use the algorithm more directly), then to find inverses for linear trans- formations, then to find determinants. We will use observations about the progress of the algorithm in proofs. Once we have a good method for solving systems of equations we can turn to the problems of finding bases and representing linear transformations with matrices, given choices of basis for domain and codomain. Good choices entail finding eigenvalues, so we’ll develop some algorithms for that problem. Eigenvalues for linear transformations are extremely important in appli- cations and as a theoretical tool. In particular, they tell us a lot about the dynamics of iteration of linear transformations. We will first gain an under- standing of the kind of information they give by looking at small examples and then see how they can be found and used in more general situations. Addition of an inner product to our structure will allow us to consider length and angle (or at least orthogonality) in our spaces. This leads to the notion of best approximation through orthogonal projection, the basis for Fourier series and wavelet analysis and several other important notions in numerical analysis. We will use Mathematica to illustrate the concepts and take the drudgery out of some of the computations. Some time will need to be spent outside of the schedulded class time to complete the lab problems.

Exams and Grading

There will be three in exams in the course. See the course calendar for tentative dates. The third is at the time scheduled for the final. Each of

include the names of all of the students who worked together in the group. If you use outside sources, cite them. If you get help from an individual, give credit. It is not wise for you to neglect learning how to do the work on your own, since exams will all require all work to be done individually. Any cheating on exams or collaboration on assignments where it has been explicitly prohibited will be treated as a violation of the policy on academic dishonesty in the student handbook and will be reported to the Associate Provost.