Math 110 Professor Ken Ribet Linear Algebra Spring 2005 January 18 2005 See http math berkeley edu ribet 110 information about for The textbook The discussion sections Exam dates Grading policy Office hours link 1 Note especially that there is an online discussion group Google s Math 110 The URL for the group is http groups beta google com group Math110 If you join the group you can post comments and questions If you don t join you can still lurk and read what other people have written 2 We have a fine lineup of experienced GSIs ChuWee Lim Scott Morrison John Voight Discussion sections are on Wednesdays see the class Web page for times and room numbers Note that 103 is in 433 Latimer not 435 Latimer Some of the sections are full If you want to change your section to one that is full you need to speak with Barbara Peavy in 967 Evans The burden will be on you to convince her that your schedule requires you to change into one of the full sections 3 An initial question for feedback Is lecturing with a laptop going to be effective Mathematicians have traditionally written on the board with railroad chalk fat chalk when giving large courses Should I do that Should I use transparencies Some combination Post to the newsgroup with your thoughts Meanwhile I will try to lecture today with a laptop and the projector 4 Even though we re in a large room don t hesitate to stop me to ask questions One good thing about the laptop is that I can see you while I m speaking You can interrupt don t be shy If you would like some clarification so will your friends 5 Vector Spaces and Fields I imagine that you know more than a bit about matrices systems of equations linear maps You have taken Math 54 In Math 110 we study vector spaces and linear transformations more abstractly We will be interested a choosing bases for vector spaces in such a way that linear transformations are represented by especially nice matrices 6 What is a field It s a set with an addition and a multiplication the system is required to satisfy a list of familiar looking axioms Appendix C of book Some examples The field R of real numbers The field C of complex numbers The field 0 1 with two elements The field Q of rational numbers 7 Fix a field F A vector space over F is a set V together with two additional structures an addition law x y 7 x y on V and an operation of F on V F V V a x 7 ax These operations satisfy a whole bunch of axioms VS 1 VS 8 in the book We refer to V with its two additional structures simply by the letter V 8 Examples An example that we all know for each n 1 the set F n c1 cn ci F for 1 i n becomes an F vector space with the operations c1 cn d1 dn c1 d1 cn dn and a c1 cn ac1 acn componentwise addition and multiplication by elements of F 9 The 0 vector space is the set V 0 with the obvious operations a 0 0 for all a F 0 0 0 It is true somehow that 0 F n when n 0 Warning I sometimes write simply 0 for the 0vector space Our authors are careful to write 0 instead I like my shorthand but don t want to force it on anyone 10 Another example a11 a21 am1 the set of m n matrices a12 a1n a22 a2n am2 amn again with component wise addition and the obvious multplication by elements of F This example is visibly a re labeling of F mn This set is called Mm n F in the book 11 We can take V to be the space of all polynominals over F of any degree or the space of polynomials of degree n for a fixed non negative integer n These spaces are called F and n F respectively Since a polynomial of degree n cnxn cn 1xn 1 c1x c0 is just a string of n 1 numbers n F is a suave way of writing F n 1 12 If S is a set we can take V to be the set of functions f S F and define the operations in a pointwise manner Thus f g is the function f g s f s g s and af is the function taking s to af s If S 1 2 n we get F n An interesting variant is to take V instead to be the set of functions S F that have only a finite number of non zero values This is a genuine variant if S is infinite 13 For example if S is the set of natural numbers i e non negative integers the variant V that we have defined is the set of sequences c0 c1 c2 such that cm 0 for m sufficiently large Such sequences are really the same thing as polynomials Hence the V that you get in this case is the F vector space of polynomials over F If you remove the requirement that cm be zero for large m you get formal power series nonterminating polynomials such as 1 x x2 x3 14 The axioms First V is an abelian group under addition We have x y y x for x y V For x y z V x y z x y z There is a unique 0 V such that x 0 x for all x V For each x V there is a unique x V such that x x 0 15 We should all know the proof that 0 is unique and that additive inverses are unique For the first assertion suppose that 0 and 00 both play the role of zero Then 0 00 is both 0 and 00 Hence 0 00 If y and z are additive inverses for x then y y 0 y x z y x z 0 z z 16 Next two axioms about the action of F on V For all x V 1 x x For a b F and x V a bx ab x Finally there are two distributive laws a x y ax ay and a b x ax bx valid for a b F x y V 17 The deal about the axioms is that you want them to be easy to check but want them also to be powerful enough to imply familiar looking statements that had better be true for us to preserve our sanity For example we want 0 x 0 and x x for all x V a x a x ax for a F x V etc etc 18 Let s prove something Take a vector x in a vector space V Consider 0 x We d be amazed if 0x turned out to be anything other than 0 the 0 vector How do we actually prove that 0x 0 19 For …