Access or timely update to this web site is not guaranteed. Attend all classes for the latest information.

Go here for IPAD/Iphone/Mac compatible videos . The following videos are in *.wmv format for Windows Media Player.

Go to The Start of Linear Algebra Lectures then scroll down to the current lecture.

1) Home work assignment

2) Course Information, Spring 2013, subject to change

4) Text: Elementary Linear Algebra with Applications, 3rd Edition, by Richard Hill, ISBN 0030103479.

Available at bookstores as well as online, such as

Amazon Barnes and Noble Ebay Abebooks

At a minimum, redo problems done in class or text, and then do the problems listed on the homework sheet given at the beginning of the semester.

Follow the methods used in class or shown in the text book.

Compare your final answer with the solution to the odd-numbered problems at the end of the text.

Form study groups.

Use free tutoring available in L209, M-R 8-6, F 8-2. Talk to several tutors. Ask questions collectively.

Bring up your un-answered questions in office hours.

Or bring up your un-answered questions in class.

Give yourself self-tests: Write problems (whose solutions you have seen) on a flash card. Indicate where solution can be found and allocated time. Store in a box. Draw 10 random questions, give yourself a timed test. Grade using the available solutions. Re-study problems you missed.

In case we use the hybrid approach you want to make sure your your lecture note and homework binder is complete, searchable and readable to yourself.

Only one binder note-book of your own hand-written items is allowed on tests.

No printed paper allowed. No copied paper allowed. This will result in a grade of F for the course even if not used!

No type of loose paper of any form is allowed on tests. No torn/glued/stapled/sewn/stitched paper allowed.

You may have a regular basic scientific calculator (about $20) on tests. Use of advanced calculators (such as TI80 etc) will result in a grade of F for the test.

No wireless devices allowed. No cell-phone/ipod-style calculator allowed.

If you do not have your own proper calculator then you will take your test without one.

Go here for IPAD/Iphone/Mac compatible videos . The following videos are in *.wmv format for Windows Media Player.

The video content of this page works with Windows and has WMV (Windows Media Video) format. In particular you should not experience any problems with Windows Media Player and a variety of browsers (Internet Explorer, Firefox, Chrome) on Windows XP/ Vista/ Windows 7. Other operating systems (e.g Apple/Mac or Unix) are not compatible with the videos without additional software.

How to view the videos on MacBook: (Suggested by Kevin Tabien Sept 2009).

1. Download MPlayer . Get the OS X 1.0 version.

2. Download DLL files . Get the file windows-all-20071007.zip ( or the most recent version).

3. Go to Finder in the tab go, click the "Go to Folder Option" and type "/usr/lib/" in here you will make a new folder called "win32". You may need to type in password for your computer and become administrator as the "usr" folder is hidden and writing or modifying files here can cause issues so only do what is needed. If you do not fee comfotable with this ask at the computer help window in Maes Building.

4. In this folder you will extract all the contents of the downloaded zip file.

5. Open MPlayer, go to Preferences, and change the video output model to Quartz/Quicktime.

6. Now open the video files using MPlayer.

Online Courses, Texts, Interactive Calculators, Simulators, Games, and Demos Related to Linear Algebra:

Linear Algebra by Gilbert Strang a complete course package. You may also use 1999 version.

Linear Algebra Course by Alex Postnikov, 2009

Matrix analysis and applied linear algebra Text by Carl D. Meyer.

Introduction to matrix algebra Text by Autar K. kaw.

Linear Algebra Text by Jim Hefferon. Large PDF Download.

How to speed up the video replay.

Interactive Apps

Linear Algebra Toolkit Calculator by Przemyslaw Bogacki, Old Dominion University

Basic Matrix algebra Calculator for multiplication, determinant, inverse, Gussian elimination or rref (row reduction to echelon form).

Step-by-Step ow Operations

3D plots

Games

Lights Out 1 , By David Guichard, Whitman College, A linear algebra game somewhat similar to the classical Othello/Reversi game, but single player. Can you find out how to solve the Lights Out puzzle by applying linear algebra?

Lights out 2, by Misha Guysinski, at Penn State.

Fifteen , by Misha Guysinski, at Penn State. This one requires permutations.

Read Section 1.1 Pages 1-3

Do problesm from 1-20.

Lecture 1, Introduction

linear equations, a solution, solving, solution set, parametric solution, system of linear equations, linear systems, inconsistent systems, consistent systems, graphical solution, a linear system may have 0 or 1 or infinitely many solutions.

Read Section 1.1 Pages 4-9

Do problems from 21-32.

Lecture 2, Introduction

Motivation for studying linear algebra, Gaussian elimination, triangular form, echeleon form, backsubstitution, coefficient matrix, augmented matrix, right hand side column, elementary operations on a linear system, elementary row operations on a matrix.

Matrix Toolkit

If you want to SEE how a certain formula with up to three variables looks use the following interactive page/applet/app:

3D plots

Read Section 1.2 Pages 10-13.

Do problesm from 1-14.

Lecture 3, Gaussian Elimination

parametric equations and deciding 0, 1, or infinitely many solutions, echelon form, leading zeros, pivot elements, leading/dependent variables, free/independent variables, how to solve an equation given in echelon matrix form.

Read Section 1.2 Pages 13-21.

Do problems from 15-29.

Lecture 4, Gaussian Elimination

Elementary row operations used to reduce an equation to its echelon form; so that it can be solved by back-substitution.

To check your echelon form or to solve your system you may try this calculator (use the 2nd or 4th tool).

Matrix Toolkit

Read Section 1.3 Pages 23-27

Do problems from 1-14.

Lecture 5, Matrix Algebra

Problem 15 from 1.2, matrix entry or element, matrix size or dimension, double subscript, matrix addition and subtraction, scalars and matrices, multiplication of a scalar and a matrix, dot product or inner product, matrix product as a collection of dot products.

To check your mutiplication try this calculator

Basic Matrix algebra

To see

demo of multiplication search for "work out", click on "multiply two matrices", follow directions.

Read Section 1.3 Pages 27-29

Do problems from 15-16.

Lecture 6, Matrix multiplication

more on matrix product as a collection of dot products, general formula using summation notation

To check your mutiplication try this calculator

Basic Matrix algebra

To see

demo of multiplication search for "work out", click on "multiply two matrices", follow directions.

Read Section 1.3 Pages 30-31

Do problems from 21-26, try 27-32, especially 31, skip 17-20.

Lecture 7, Interpretations of Matrix multiplication

multiplication by a diagonal matrix, a linear system written as matrix product, substitution as matrix multiplication, matrix multiplication as a sum of products of columns with rows.

Read Section 1.3 Pages 32-34, 1.4 Pages 38-39

Do problems from 1.3, 33-42. At least one from each type even though not on the blue page list!

Lecture 8, Different styles for Matrix multiplication

Interpretations of A.B in terms of dot products of rows of A with columns of B, A with columns of B, rows of A with B, row matrices of A with column matrices of B, sum of outer products of columns of A with corresponding rows of B. AX, X a vector, is the linear combination of columns of A with weights same as entries of vector X.

Read Section 1.4 Pages 38-45

Do problems from 1,3,5,8,10,11,14.

Lecture 9, Inverses and elementary matrices

Elemetary matrix associated with multiplication of a row and exchange of two rows.

Read Section 1.4 Pages 38-45

Do problems from 1-24, 36, 39, 40, 41-43, 45, 49, 50 .

Lecture 10, Inverses and elementary matrices

Elementary COLUMN operations (related to problem 50).

Problem 37 from 1.3, different styles (row expansion, column expansion, outer product expansion) for multiplication of matrices, related to lecure 8.

Elemetary matrix associated with adding multiple of a row to another row.

General procedure for finding the inverse.

Read Section 1.4 Pages 48-49

Do problems from 9-20, 25-45.

Lecture 11, Finding Inverses using elementary matrices

An example of 3*3 matrix being inverted. Short discussion of non-invertible matrices.

Read Section 1.5 Pages 57-64

Do problems from 1-8.

Lecture 12, LU factorization

LU factorization for a simplae case (without permutatio), description of algorithm and justification of procedure.

Read Section 1.5 Pages 62-64.

Do problems from 9-24.

Lecture 13, Solving AX=B via LU factorization

Advantages of LU factorization, solving systems with factorization, general permutation matrices, PA=LU.

Read Section 1.5 Pages 64-67, 1.6 Pages 71-72, 3.1 Pages 130-131

Do problems from 1.5 33-36, from 1.6 1-7, 13,14,19,20.

Lecture 14, Solving AX=B via PA=LU factorization,

an example, symmetric, skew-symmteric, determinants of 2*2 and 3*3 matrices, vectors in physics

Read Section 3.1 Pages 132-137, 151

Do problems from 3.1, 1-30 (postpone "span" questions)

Lecture 15, Vectors

scalars, vectors, matrices, graphical addition, subtraction, and scalar multiplication of vectors

Read Section 3.1 Pages 132-141

Do problems from 3.1, 1-36

Lecture 16, length and dot product of vectors

span, distance, norm (length, magnitude, size) of a vector, angle between vectors, dot product through angle, length interm of dot product, dot product in terms of components, example: find the angle between two vectors.

Read Section 3.1 Pages 140-144

Do problems from 3.1, 31-42

Lecture 17, Dot Product theorem, Projections

Example 5,11,36. Proof of Law of Cosines, Proof of U.V= u1 v1+ u2 v2 + u3 v3 = |U| |V| cos of angle. Formula for projections

Read Section 3.1 Pages 144-146

Do problems from 3.1, 43-50

Lecture 18, Gram-Schmidt Process

Example 21. Producing a set of mutually perpendicular vectors P_i out an arbitrary given Vectors V_i using projections.

Read Section 3.2 and 3.3 Pages 149-165

Do problems from 3.2, 3.3 all

Lecture 19, Euclidean n-Space, General Vector Spaces

Closure.

Review Section 1.4,1.5,1.6,3.1,3.2

Lecture 20, Review of Inverse, LU, PA=LU

A biref review.

Read Section 3.4 Pages 165-169

Do problems from 3.4, 1-24

Lecture 21, Subspaces

Closure with respect to vector addition and scalar multiplication.

Read Section 3.4 Pages 168-169

Do problems from 3.4, 1-24

Lecture 22, Subspaces

examples of subspaces

Read Section 3.4 Pages 169-173

Do problems from 3.4, 33-40

Lecture 23, Subspaces

Null spaces, NS(A), linear combinations

Read Section 3.4 Pages 173-176, 3.5 179-180

Do problems from 3.4, 41-48, 3.5 1-3

Lecture 24, Subspaces

linear combinations, interpretation of AX=B, linear dependence and independence

Read Section 3.5 Pages 179-180

Do problems from 3.5 1-6

Lecture 25, Linear Independence

Problems 29 and 38 from 3.4, Intro to linear dependence and independence

Read Section 3.5 Pages 180-185

Do problems from 3.5 7-24

Lecture 26, Linear Independence

linear dependence and independence

Read Section 3.6 Pages 187-192

Do problems from 3.6 1-20

Lecture 27, Basis

Given a set of vectors how do we detect if they are a basis for a given space?

Read Section 5.1 Pages 320-326

Do problems from 5.1 1-20

Lecture 28, Determinants

Introduction to determinants, short cuts, expansion/recursive formula

Read Section 5.1 Pages 326-329

Do problems from 5.1 21-39

Lecture 29, Determinants

Examples, three theorems.

Read Section 5.2 Pages 331-334

Do problems from 5.2 1-9, part b only of 11-22

Lecture 30, Introduction to Eigenvalues and Eigenvectors

Examples, Definitions.

Read Section 5.2 Pages 331-335

Do problems from 5.2 1-9, only part a,b of 11-22

Lecture 31, real, complex, double eigenvalues, characteristic polynomial of a matrix

several examples.

Read Section 5.2 Pages 331-335

Do problems from 5.2 1-9, only part a,b,c of 11-22

Lecture 32, Finding eigenvectors of a matrix, matrix factorization into eigenvector matrix *eigenvalue matrix* inverse of eigenvector matrix

several examples.

Read Section 5.2 Pages 331-335

Do problems from 5.2 18

Lecture 33, Finding eigenvectors of a 3*3 matrix,

One example.

Read Section 5.2 Pages 331-335

Do problems from 5.2 17-22

Lecture 34, Finding eigenvectors of a 3*3 matrix, diagonalization: matrix factorization into eigenvector matrix *eigenvalue matrix* inverse of eigenvector matrix f(A)= V f(lambda) V^ -1, several examples.

Read Section 5.2 Pages 331-335

Do problems from 5.2 17-22

Lecture 35, Repeated eigenvalues Finding eigenvectors and basis of eigenspaces for repeated roots of characteristic polynomial, several examples.

Read Section 5.2, Pages 338-339, 5.3 342-349

Do problems from 5.2 part (f) of 11-22, 5.3 1-26

Lecture 36, trace, determinants, and eigenvalues, diagonalizability sum of eigenvalues=trace, product of eigenvalues=determinant, f(A)=V f(lambda) V^-1

Read Section 5.3 342-349

Do problems from 5.3 1-26

Lecture 37, Diagonalization

Examples, non-diagonalizable matrices.

Read Section 5.4 352-358

Do problems from 5.4 1-9

Lecture 38, Symmetric Matrices

Properties, Orthogonal (Orthonormal) matrices, Theorem: Eigenvalues of a real symmetric 2*2 matrix are real.

Read Section 5.4 352-358

Do problems from 5.4 1-9

Lecture 39, Review of Symmetric Matrices

Solution of problem 1

Lecture 40, Partial Review of eigenvalues, eigenvectors, diagonalization, solution of a system of diff eq Covers parts of 5.2,5.3,5.6 (for 5.6 use lecture note)

Read second part of Section 5.5 352-358

Do problems from video

Lecture 41, Markov Processes, an example