Bootcamp2011algebra

# Bootcamp2011algebra - Department of Biostatistics...

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Unformatted text preview: Department of Biostatistics Department of Stat. and OR Refresher course, Summer 2011 Linear Algebra Original Author: Oleg Mayba (UC Berkeley, 2006) Modified By: Eric Lock (UNC, 2010 & 2011) Instructor: Eric Lock (UNC at Chapel Hill) Based on the NSF sponsored (DMS Grant No 0130526) VIGRE Boot camp lecture notes in the Department of Statistics, University of California, Berkeley June 7, 2011 Contents 1 Introduction 2 2 Vector Spaces 2 2.1 Basic Concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2.2 Orthogonality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.3 Gram-Schmidt Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 3 Matrices and Matrix Algebra 8 3.1 Matrix Operations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 3.2 Special Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 3.3 Fundamental Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 4 Least Squares Estimation 13 4.1 Projections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 4.2 Applications to Statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 5 Differentiation 16 5.1 Basics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 5.2 Jacobian and Chain Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 6 Matrix Decompositions 19 6.1 Determinants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 6.2 Eigenvalues and Eigenvectors . . . . . . . . . . . . . . . . . . . . . . . . . . 21 6.3 Complex Matrices and Basic Results . . . . . . . . . . . . . . . . . . . . . . 23 6.4 SVD and Pseudo-inverse . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 7 Statistics: Random Variables 28 7.1 Expectation, Variance and Covariance . . . . . . . . . . . . . . . . . . . . . 28 7.2 Distribution of Functions of Random Variables . . . . . . . . . . . . . . . . . 30 7.3 Derivation of Common Univariate Distributions . . . . . . . . . . . . . . . . 33 7.4 Random Vectors: Expectation and Variance . . . . . . . . . . . . . . . . . . 36 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 8 Further Applications to Statistics: Normal Theory and F-test 38 8.1 Bivariate Normal Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . 38 8.2 F-test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 9 References 41 1 1...
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## This note was uploaded on 11/17/2011 for the course STOR 664 taught by Professor Staff during the Fall '11 term at UNC.

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