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# Linear Algebra MATH 110

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This 19 page Class Notes was uploaded by Kavon Feest on Thursday October 22, 2015. The Class Notes belongs to MATH 110 at University of California - Berkeley taught by K. Ribet in Fall. Since its upload, it has received 17 views. For similar materials see /class/226584/math-110-university-of-california-berkeley in Mathematics (M) at University of California - Berkeley.

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Date Created: 10/22/15

Math 110 Professor Ken Ribet Linear Algebra Sp ng2005 January 18 2005 See httpmathberke1eyeduquotribet110 for information about a The textbook 0 The discussion sections 0 Exam dates 0 Grading policy 0 Office hours link Note especially that there is an online discussion group Google s Math 110 The URL for the group is httpgroupsbetagooglecomgroupMath110 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 We have a fine lineup of experienced GSls Chu Wee 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 ls 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 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 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 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 a The field R of real numbers a The field C of complex numbers a The field 01 with two elements a The field Q of rational numbers Fix a field F A vector space over F is a set V together with two additional structures an addition law my I gt m 3 on V and an operation of F on V FgtltV gtV ami gta 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 Examples An example that we all know for each n 2 1 the set Fquot clcnici E Ffor 1 g 239 g n becomes an F vector space with the operations C17Cnd1dnC1d1Cnl dn and a 01cn aclacn componentwise addition and multiplication by elements of The O vector space is the set V 0 with the obvious operations aO 0 for all a E F 00 0 It is true somehow that 0 Fquot when n 0 Warning I sometimes write simply O for the 0 vector 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 the set of m X n matrices Q11 Q12 39 39 39 am Q21 Q22 39 39 39 272 7 aml am2 39 39 39 amn again with component wise addition and the obvious multplication by elements of F This example is visibly a re labeling of Fmquot This set is called menF 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 3 n for a fixed non negative integer n These spaces are called and respectively Since a polynomial of degree 3 n cnmquot cn1mn1 011 CO is just a string of n 1 numbers is a suave way of writing F71 12 If S is a set we can take V to be the set of functions f S gt F and define the operations in a pointwise manner Thus f g is the function f gs fs and of is the function taking 3 to afs If S 12 n we get Fquot An interesting variant is to take V instead to be the set of functions S gt 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 ie non negative integers the variant V that we have defined is the set of sequences C0 0102 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 non terminating polynomials such as 1mm2m3 14 The axioms First V is an abelian group under addition a We havemyym for 133 E V o For myz V myz myz c There is a unique 0 E V such that 70 0 m for all m E V o For each m E V there is a unique m E V such that m 96 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 0 both play the role of zero Then 0 0 is both 0 and 0 Hence 0 0 If y and z are additive inverses for ac then yy0ymzymz0zz 16 Next two axioms about the action of F on V o ForallmEV1mm o For ab E F and m E V abm 01996 Finally there are two distributive laws amy am ay and a 990 am bx valid for a b E F 90 y E V 17

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