The University of Science and Technology. And Linear Algebra.

MA 2071 Linear Algebra B '03

Instructional Staff

instructor: John Goulet Stratton 201 A goulet@wpi.edu

PLAs: may emailed via linalgpla@wpi.edu

Quizzes:

6) Friday, Dec 12 Diagonalization

basic definition of what eigenvalues and eigenvectors are ( Ax = λx , L(x) = λx )

computing eigenvectors given eigenvalues

deciding if a matrix can be “diagonalized” (finding a basis of independent eigenvectors)

finding matrices P and D such that A = PDP^-1

sample quiz 6 here with solutions here

yes, there is a review session as usual on Thursday night. (7-8:30 in Stratton 309)

Final Exam - Thursday in class

actual exam will appear here this weekend

Text: Lay Linear Algebra and It's Applications

Grade:

quizzes (6) 40%

projects 30%

final exam 20%

homework 10%

5 point bonus for perfect attendance!

Homework

assignments given daily at the beginning of class, by section

see the following file

Course Philosophy

follow this link for an overview of what this course is about

a paper discussing the evolution of this course is at this link.

Core Material Covered:

Linear Systems

types of problems

types of solutions

arbitrary variables

the Gauss-Jordan algorithm

Matrix Arithmetic and Algebra

matrix arithmetic

matrix algebra

diagonal and symmetric matrices

A = PDP^-1

inverses and powers of matrices

relation of matrices to solutions of systems – rank

change of coordinates

Row vs Column Views of Solutions of Systems

Ax thought of as a linear combination of the columns of A

Vector Spaces

basis

concept of building blocks

determining if a set is a basis or not

coordinates and coordinate changes x = Pc

how to build and use the P matrix for coordinate changes

orthogonal basis

basis where all vectors in it are perpendicular to each other (0 dot product)

at the heart of Fourier analysis

key formula: c_i = xv_i/v_iv_i

special case: orthonormal basis (all unit vectors)

real and abstract vector spaces

“closure” requirements

Rⁿ

planes and lines through the origin

solutions to homogeneous systems

solutions to homogeneous 2^nd order differential equations

sets of functions

Determinants

Basic Properties

Relation to inverses, to solutions of systems, to rank

Useful Theorems

Product Rule, Det(A^t) = Det(A) etc

Computing by hand:

XXX rule, Cofactor Expansion

Cross Products

Geometric Interpretation

Linear Transformations

definition

Laplace and Fourier; derivatives and integrals

matrix of

application to rotations, reflections

determined entirely by effect on basis for domain

when is a linear transformation 1-1??

Diagonalization and the Principal Axis Theorem

eigenvalues and eigenvectors – recognizing, computing

when can a matrix be diagonalized?

how do you diagonalize?

applications:

powers of matrices

behavior of dynamical systems

conic sections

Technology

Using Maple for linear algebra is at this link

To see notes on Matlab for linear algebra (esp eigenvalues), follow this link

Projects -

A primary component of the WPI approach to education is the project. As all of you know,

you will participate in team projects at junior and senior level (the IQP and MQP, respectively).

These will require many things of you: working in a team, communicating orally and in writing,

planning, using technology and other resources, learning things you knew nothing about before,

and finally, providing a solution to a complex problem of society or in one’s major.

If you are going to be successful in projects at the junior and senior levels, you will need some experience prior to that. That is one of the goals of this course.

follow this link for Project #3 (this supercedes and replaces any other "project 3"s listed earlier)

due Monday, Dec 15 in class

Projects by Major:

Bio projects: follow this link

Civil Engineering: follow this link.

Computer Science: follow this link.

ECE majors: follow this link.

Mechanical Engineering Majors: follow this link.

Physics Majors: follow this link.