Math 110 Fall 05 Lectures notes 1 Aug 29 Monday Name class URL www cs berkeley edu demmel ma110 on board See Barbara Peavy in 967 Evans Hall for all enrollment issues All course material will be on the web page If this is inadequate please send me email and I will put copies at Copy Central Northside Read Course Outline on web page for course rules and grading policies You are responsible for reading this and knowing the rules Text which we will follow fairly closely Linear Algebra 4th Edition by Friedberg Insel Spence same as last semester Topics following table of contents Chap 1 Vector spaces vectors and their basic properties Chap 2 Linear transformations matrices and their basic properties Chap 3 Elementary Matrix Operations quickly Chap 4 Determinants Chap 5 Diagonalization eigenvalues and vectors Chap 6 Inner products orthgonality singular value decomposition Chap 7 Jordan Form generalizations of eigenvectors Examples why is linear algebra important Linear Equations Consider new Bay Bridge how is it designed How do they know it will be strong enough to hold up the traffic If you imagine a car sitting on the end of a beam you can write a simple relationship F force from weight of car on bridge k stiffness of beam x how much beam bends So if you know the weight of the car and the stiffness of the beam you can solve one linear equation in one unknown F kx for x F k to get the amount the bridge bends and compare that to how much you are willing to let it bend In a real bridge there are many cars many beams and many x s describing how each beam bends and the the beams are connected So instead of of 1 linear equation in 1 unknown you get thousands of linear equations in thousands of unknowns which civil engineers have to solve The same process applies to any important structures buildings airplanes cars certain computer chips See CE 130 ME 104 1 All radio light and other electromagnetic radiation in free space satisfy a system of linear equations called Maxwell s equations These are partial differential equations but linear nonetheless See Ph 7B or Ph 110 EG Aim 2 flashlights at board and observe that lit spot 1 is same whether or not beam 2 passes through beam 1 ASK WAIT Can you explain this using linear algebra Eigenvalues and Eigenvectors ASK WAIT Each of you depends on an eigenvector of one of the world s largest matrices many times each day How big is the matrix Hint what number do you see when you go to www google com Schroedinger equation every atom molecule your physical body is most acccurately described as an eigenvector of Schroedinger s equation which is a partial differential equation The eigenvalues correspond to energy levels and the eigenvectors describe how the electrons are distributed around all the atomic nuclei See Ph 7C or Ph 137 What we will cover We will concentrate on definitions and theorems describing basic properties of these linear algebra objects like linear transformations their inverses when they exist eigenvalues and eigenvectors when they exist In particular you will practice reading and writing clear and correct mathematical proofs We will also try to look at concepts and problem solving from at least 2 points of view algebra and geometry because linear algebra natually incorporates and can be understood both ways EG One can either say 2 equations in 2 unknowns can or 2 lines in the plane can 1 have a unique solution 1 intersect in a point 2 have no solutions 2 be parallel and not intersect 3 have infinitely many solutions 3 be identical What we will not cover practical algorithms for solving linear equations finding eigenvalues vectors Ma128ab Ma221 This course is a prerequisite for such courses Prereq Math 54 including definitions of sets functions Apps A B 2 Homework will be due Thursdays at the start of section There will be brief weekly quizzes to make sure people are keeping up To get started let s talk carefully about the numbers that we will be writing in our vectors and matrices As motivation Consider solving 2 x 1 even though there are only integers in the equation we can t solve for x if we only look for x in Z set of integers we need Q set of rational numbers Consider solving x 2 2 0 even though there are only rational numbers in the equation we can t solve for rational x we need real numbers R Consider solving x 2 pi 0 even though there are only real numbers in the equation we can t solve for real x we need complex numbers C DEF A set of numbers also called scalars F is called a field if there are two binary operations addition and multiplication so that x y and x y are unique numbers in F for all x and y in F and such that and satisfy the following conditions 1 for all x and y in F x y y x and x y y x commutativity of addition and multiplication 2 for all x y z in F x y z x y z and x y z x y z associativity of addition and multiplication 3 There exist distinct scalars 0 and 1 such that for all x in F x 0 x and x 1 x existence of identity elements for addition multiplication 4 For all x in F and nonzero y in F there exist x and y such that x x 0 and y y 1 We denote x by x and y by y 1 existence of inverses for addition multiplication 5 for all x y z in F x y z x y x z distributivity ASK WAIT Is Z a field Why ASK WAIT Is Q a field ASK WAIT Is R a field ASK WAIT Is C a field ASK WAIT Is S q1 q2 sqrt 2 q1 and q2 in Q a field ASK WAIT Is A all roots of polynomials with integer coeffs a field 3 EG Z2 0 1 with 0 0 0 0 1 1 1 1 0 0 0 0 0 1 0 1 1 1 Can show this is a field homework ASK WAIT Does this field with its operations have other names EG Zp 0 1 2 p 1 where p is a prime x y x y mod p remainder after dividing x y by p x y x y mod p remainder after dividing x y by p Can show this is a field homework Used in cryptography see Ma55 DEF A field F has characteristic p if 1 1 1 p times 0 for some positive integer p Otherwise F has characteristic 0 ASK ASK ASK ASK WAIT WAIT WAIT WAIT What What What What is is is is the the the the characteristic characteristic characteristic characteristic of of of of Q R C S A Z2 Zp EG F x rational functions in x with coefficients from another field F of characteristic 0 So …