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# Linear Algebra MATH 110

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This 7 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 Staff in Fall. Since its upload, it has received 32 views. For similar materials see /class/226605/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 Fall 05 Lectures notes 8 Sep 16 Friday Homework due Thursday Sep 22 Sec 21 1 justify your answers 3 4 5 6 7 13 14 15 16 17 18 19 20 21 22 25 35 38 Start reading chapter 2 Just as Chapter 1 generalized ideas about vectors familiar from math 54 Chapter 2 will generalize ideas about matrices Ma54 gt Ma110 real numbers gt fields vectors of real numbers gt vector spaces EX X1X2 gt also PnF FuncRR etc lines and planes through Ogt subspaces EX aX1X2 gt spans1s2sn lines are 1D planes 2D gt dimension of any vector space matrices gt linear transformations EX 1 2 gt also differentiation integration etc 3 4 multiplying matriXvector gt applying a linear transformation to a vector EX 1 2 X1 X12X2 gt also Tfx f x 34 k2 Bm1m multiplying matriXmatriX gt composing linear transformations forming invQMQ gt changing the basis of a linear transformation Def Let V and W be vector spaces over F A function T V gt W is a linear transformation from V to W or just linear if for all Xy in V and c in F a TXy TX Ty b TcX cTX note a gt b if F Q but in general need both Lemma TOV UW T linear if and only if TcXy cTX Ty T linear implies TX yTX Ty T linear if and only if Tsumi civi sumi ciTvi EX V R 2 W R TXy x2y ASK amp WAIT Why is T linear EX V W R 2 TXy Xy 3X2y ASK amp WAIT Why is T linear This is also written as the matriX vector muliplication TXl1llxl xy yl 3 2 y 3x2yl Indeed T F n gt F m where T is multiplying an n vector by an m X n matrix to get an m vector is a linear transformation In section 22 we will systematically write all linear transformations from F n to F m as multiplying an n vector by an m X n matrix to get an m vector But these are not the only linear transformations EX V differentiable functions from 01 to R W functions from 01 to R TfX f X integralO X t 2ft dt ASK amp WAIT Why is T linear Is T V gt V EX V W R 2 TthetaXy vector Xy rotated clockwise by theta ASK amp WAIT Why is T linear Def V W TX X called identity transformation written IV or I Def TX OW called zero transformation Def Let T V gt W be linear Then the null space of T NT is the set of all vectors v in V such that Tv OW The range space or just range of T RT is the set of all vectors w Tv for all v in V Thm 1 NT and RT are subspaces of V and W resp Proof Consider NT clearly OV in NT since TOV OW Since T is linear v1 and v2 in NT implies Tav1 bv2 Tav1Tbv2 aTv1 bTv2 aOV bOV OV so NT is closed under and and so a subspace NeXt consider RT clearly OW in RT since TOV OW If w1 and w2 in RT then there are v1 and v2 so that w1Tv1 and w2Tv2 and so aw1bw2 aTv1bTv2 Tav1Tbv2 Tav1bv2 is in RT too so RT is closed under and and so a subspace EX V R 2 W R TXy X2y Xy X2y O f2 1 for all reals f RT w wx2y for some Xy R ASK amp WAIT Why is RT R EX V W R 2 TXy Xy 3X2y NT Xy XyO and 3X 2y O solving these 2 equations in 2 unknowns gt so NT OV RT wz w Xy and z 3X2y solving these 2 equations in 2 unknowns and X X y o for some Xy gt y 15z 35w 15Z 25w so any wz in range Thm 2 If S v1vn is a basis for V then RT spanTv1Tvn Proof if S is a basis all v in V are of the form sumi aivi so all w in RT are of the form Tsumi aivi sumi Taivi sumi aiTvi so Tv1Tvn is a spanning set ASK amp WAIT ls Tv1Tvn necessarily a basis EX V R 2 W R 3 TXy xy 3x2y X 4y To compute NT note XyO and 3X2y0 implies XyO from before so NT OV as before RT spanT1O TO1 span131124 Let T V gt W be a linear transformation with null space NT and range RT If NT is finite dimensional we call its dimension the nullity of T written quotnullityTquot If RT is finite dimensional we call its dimension the rank of T written quotrankTquot Thm 3 Dimension Theorem Let T V gt W be a linear transformation If V is finite dimensional then dimV rankT nullityT EX Just as in chapter 1 where we asked what our theorems about vector spaces meant for R 3 or R n in chapter 2 we can ask what our theorems about linear transformation mean for matrices especially simple matrices like diagonal ones Consider V R n W R m and T is multiplication by an m X n matrix Suppose T is diagonal with Os and 1s on the diagonal ie only some Tii can be nonzero such as T 1 O O O we will show that rankT nonzero columns of T O 1 O O nullityT zero columns of T 0000 so rankT nullityT nonzero columns of T zero columns of T columns of T n dimV as claimed by Dimension Theorem For simplicity we suppose T11 T22 Trr 1 and the rest are 0 So T has r nonzero columns and n r zero columns Then it is easy to see TX TE X1 E l E l E l E xr Exrl E xr1 E l E l E l E Xn E O l where there is one zero in the result vector for every zero row in T So we can see that that RT is the space spanned by all vectors of the form on the right above which has dimension r nonzero columns of T and NT is all vectors of the form O O Xr1 Xn ie a space of dimension n r zero columns of T as desired there are m r zeros Proof of Dimension Theorem Since NT is a suspace of the finite dimensional space V it is also finite dimensionsal and has a basis v1vk This basis can be extended to a basis of V Replacement Theorem call it v1vn We claim Tvk1Tvn is a basis of RT Assuming this for a moment since it contains n k vectors we get dimV n k n k dimNT dimRT as desired To see Tvk1Tvn is a basis we have to show it spans RT and is independent But since v1vn is a basis of V RT spanTv1TvkTvk1Tvn span OW OW Tvk1Tvn spanTvk1Tvn so it spans RT To see it is independent suppose it is not and seek a contradiction Write OW sumik1 to n aiTvi where not all ai O T sumik1 to n aivi since T is linear so sumik1 to n aivi is a vector in NT ie sumik1 to n aivi sumj1 to k bjvj where at least one coefficient ai or bj is nonzero But this contradicts the independence of the basis v1vn Natural questions to ask about any function T not just linear ones are 1 Is T one to one ie does TXTy imply Xy 2 Is T onto ie for all w in W is there a v in V such that wTV These are important ideas because T is one to one and onto if and only if T is invertible ie an inverse function invT W gt V exists These are easy questions to answer for linear T given the rank and nullity Thm Let T V gt W where V and W are finite dimensional Then 1 T is one to one if and only if nullityT O ie NT OV 2 T is onto if and only if rankT dimW In particular since rank nullity dimV T is invertible if and only if nullityTO and dimVdimW EX Consider T F n gt F m to be multiplication by an m X n matrix where T is diagonal with all Tii 1 We ask whether T is one to one andor onto There are 3 cases depending on m and n 1 m lt n Then TX1Xn X1Xm So RT F m and T is onto But TOOXm1xn m leading zeros OW so nullityT n m gt O and T is not one to one We will see that no T can be one to one if m lt n 2 m n So T is onto for the same reason as above and TX1Xn X1Xn OW if and only if XOV so nullityTO and so T is one to one T is the identify function 3 m gt n Then TX1Xn X1Xn00 m n trailing zeros So rankT n lt m rankW and T is not onto But TvOW only if vOV so nullityT O and T is one to one We will see that no T can be onto if m gt n Proof 1 Suppose NT OV Then TXTy gt TX yO since T is linear gt X yOV since X y in NT OV so T is one to one Conversely T one to one gt TvOW only if vOV gt NT OV 2 T onto gt RT W gt rankT dimW Conversely rankT dimRT dimW so since RT is a subspace of W we must have RT W ie T is onto Corollary Suppose T V gt W and dimV dimW is finite Then the following are equivalent 1 T is one to one 2 T is onto 3 T is invertible 4 rankT dimV Proof To prove them quotequivalentquot we need to show that if any one of them is true then all of them are true To do this we will prove that 1 ltgt 2 1 and 2 ltgt 3 and 4 ltgt 1 1 ltgt 2 because 1 gt nullityTO gt rankT dimV nullityT dimV dimW gt 2 now note that all the implications work in the opposite direction too 1 and 2 ltgt 3 by the definition of invertibility 1 ltgt nullityTO ltgt rankT dimV nullityT dimV We need that the dimensions are finite for this to be true as you will see on homework EX T P2R gt R 3 is defined by Ta2X 2 a1X a0 a2a1a1aOaO Now dimP2R dimR 3 3 so we can apply the Corollary Now a2a1a1aOaO000 implies aOO gt 0 a1aO a1 gt 0 a2a1 a2 so T is one to one and hence is onto and invertible ASK amp WAIT What is its inverse The next theorem says that a linear transformation T is uniquely determined if we know what it does to a basis Thm V W be vectors spaces over F and let v1vn be a basis for V Given any subset w1wn of n vectors from W there is exactly one linear transformation T V gt W such that Tvi wi Proof Since v1vn is a basis for any v in V there is a unique linear combination v sumi1 to n aivi We can then define Tv sumi1 to n aiwi We need to prove 3 things about T 1 T is linear T vb va Tvb Tva because Tsumi bivi sumi aivi T sumi biaivi sumi biaiwi sumi biwi sumi aiwi T sumi bivi Tsumi aivi Tc vb cTvb because Tc sumi bivi T sumi cbivi sumi cbiwi c sumi biwi c T sumi bivi Tsumi aivi where ak1 and the rest are zero where ak1 and the rest are zero W also satisfies Uvi get wi then 2 TVk wk because Tvk sumi aiwi wk 3 T is unique because if U V gt for all v sumi aivi we UV U sumi aivi sumi Uaivi sumi aiUvi sumi aiwi sumi aiTvi sumi Taivi T sumi aivi T v by by linearity of U linearity of U by def of UVi by def of TVi y linearity of T by linearity of T 0quot

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