# MATH 20F Notes (Pauls Online Math Notes) - LINEAR ALGEBRA...

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LINEAR ALGEBRAPaul Dawkins
Linear Algebra© 2007 Paul DawkinsiTable of ContentsPreface.............................................................................................................................................iiOutline............................................................................................................................................iiiSystems of Equations and Matrices..............................................................................................1Introduction................................................................................................................................................1Systems of Equations.................................................................................................................................3Solving Systems of Equations..................................................................................................................15Matrices....................................................................................................................................................27Matrix Arithmetic & Operations..............................................................................................................33Properties of Matrix Arithmetic and the Transpose.................................................................................45Inverse Matrices and Elementary Matrices..............................................................................................50Finding Inverse Matrices..........................................................................................................................59Special Matrices.......................................................................................................................................68LU-Decomposition...................................................................................................................................75Systems Revisited....................................................................................................................................81Determinants................................................................................................................................90Introduction..............................................................................................................................................90The Determinant Function.......................................................................................................................91Properties of Determinants.....................................................................................................................100The Method of Cofactors.......................................................................................................................107Using Row Reduction To Compute Determinants.................................................................................115Cramer’s Rule........................................................................................................................................122Euclideann-Space......................................................................................................................125Introduction............................................................................................................................................125Vectors...................................................................................................................................................126Dot Product & Cross Product.................................................................................................................140Euclidean n-Space..................................................................................................................................154Linear Transformations..........................................................................................................................163Examples of Linear Transformations.....................................................................................................173Vector Spaces.............................................................................................................................181Introduction............................................................................................................................................181Vector Spaces.........................................................................................................................................183Subspaces...............................................................................................................................................193Span........................................................................................................................................................203Linear Independence..............................................................................................................................212Basis and Dimension..............................................................................................................................223Change of Basis......................................................................................................................................239Fundamental Subspaces.........................................................................................................................252Inner Product Spaces..............................................................................................................................263Orthonormal Basis.................................................................................................................................271Least Squares.........................................................................................................................................283QR-Decomposition................................................................................................................................291Orthogonal Matrices...............................................................................................................................300Eigenvalues and Eigenvectors...................................................................................................306Introduction............................................................................................................................................306Review of Determinants.........................................................................................................................307Eigenvalues and Eigenvectors................................................................................................................316Diagonalization......................................................................................................................................332
Linear Algebra© 2007 Paul DawkinsiiPrefaceHere are my online notes for my Linear Algebra course that I teach here at Lamar University.Despite the fact that these are my “class notes”, they should be accessible to anyone wanting tolearn Linear Algebra or needing a refresher.These notes do assume that the reader has a good working knowledge of basic Algebra.This setof notes is fairly self contained but there is enough Algebra type problems (arithmetic andoccasionally solving equations) that can show up that not having a good background in Algebracan cause the occasional problem.Here are a couple of warnings to my students who may be here to get a copy of what happened ona day that you missed.1.Because I wanted to make this a fairly complete set of notes for anyone wanting to learnLinear Algebra I have included some material that I do not usually have time to cover inclass and because this changes from semester to semester it is not noted here.You willneed to find one of your fellow class mates to see if there is something in these notes thatwasn’t covered in class.2.In general I try to work problems in class that are different from my notes.However,with a Linear Algebra course while I can make up the problems off the top of my headthere is no guarantee that they will work out nicely or the way I want them to.So,because of that my class work will tend to follow these notes fairly close as far as workedproblems go.With that being said I will, on occasion, work problems off the top of myhead when I can to provide more examples than just those in my notes.Also, I oftendon’t have time in class to work all of the problems in the notes and so you will find thatsome sections contain problems that weren’t worked in class due to time restrictions.

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