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Math 108A HW
创建日期：2024-04-01 14:33:20
所属分类：数学mathematics
标签： Math 108A
Math

Math 108A HW

Proof Problems Your answers to the problems in this section should be proofs, unless otherwise stated. F is a field, V and W are vector spaces over F. 1) Proposition 14.4 of the Lecture Notes is false when V is an infinite dimen￾sional vector space! Consider the following example: Let V = {(a1, a2, · · · , an, · · ·) ∈ F∞ | ai = 0 for all but finitely many i} be the vector space of eventually-zero sequences. Let ei ∈ V be the sequence with a 1 in position i and 0 everywhere else. Define fi : V → F by setting fi(ej ) = δij . Thus γ∗ = (f1, f2, · · · , fn, · · ·) is the “dual basis” of γ. a) Prove that γ = (e1, e2, · · · , en, · · ·) is an ordered basis for V . b) Prove that γ∗ is linearly independent in V ∗ but does not span V ∗ . (In other words, γ∗ is not actually a basis for V ∗ !) 2) Let V and W be finite-dimensional vector spaces. Let T : V → W be a linear transformation. a) Prove that T is surjective if and only if T∗ is injective. b) Prove that T is injective if and only if T∗ is surjective. 3) Let T : V → W be a linear transformation. Recall the evaluation maps ev : V → V ∗∗ and ev : W → W∗∗ defined in Theorem 14.11 of the Lecture Notes. Prove that the following diagram commutes: 1 V W V ∗∗ W∗∗ T ev ev T ∗∗ That is, prove that ev ◦ T = T ∗∗ ◦ ev. 4) Let V be a finite-dimensional vector space and let W ⊆ V be a subspace. Consider (W◦)◦ ⊆ V ∗∗, i.e. (W◦)◦ = {α ∈ V ∗∗ | ∀f ∈ W◦ , α(f) = 0}. Let ev : V → V ∗∗ be the evaluation map. Prove that ev(W) = (W◦)◦. 5) Let T : V → W be a linear transformation. Prove that the map T : V/ker(T) ∼−→ im(T) given by T(v + ker(T)) = T(v) is well-defined and an isomorphism. 6) Let T : V → V be a linear operator. Let W ⊆ V be a subspace. We say W is T-invariant if T(w) ∈ W for all w ∈ W. a) Prove that im(T) and ker(T) are T-invariant. b) Prove that W is T-invariant if and only if the induced map T : V/W → V/W given by T(v + W) = T(v) + W is well-defined. 7) Let T : V → V be a linear operator and let W ⊆ V be a T-invariant sub￾space. Let T |W : W → W denote the restriction of T to W. Prove that i) ker(T |W ) = ker(T) ∩ W ii) im(T |W ) = T(W) 8) Let W be a subspace of a vector space V . Let β = {wi}i∈I be a (possibly infinite) basis of W. Let γ = {vj}j∈J be a disjoint linearly independent set such that β ∪ γ is a basis for V . Let π : V → V/W be the quotient map. a) Prove that π(γ) = {vj + W}j∈J is a basis for V/W. b) Prove that V ∼= W × V/W. 9) Let dim(V ) = n, and let T : V → V be a linear operator. Let W ⊆ V be a T-invariant subspace of dimension k. Let γ1 = (w1, · · · , wk) be an ordered basis for W. Let γ2 = (vk+1, · · · , vn) be a disjoint linearly independent set such that β = (w1, · · · , wk, vk+1, · · · , vn) is an ordered basis of V . Let T |W : W → W denote the restriction of T to W, and let T : V/W → V/W be the induced map defined in Problem 6. Let π : V → V/W; recall from 2 Problem 8 that π(γ2) is an ordered basis of V/W. Write the n × n matrix [T] β β in block form: [T] β β = A B C D Here A is a k×k matrix, B is k×(n k), C is (n k)×k and D is (n k)×(n k). Prove that: i) A = [T|W ] γ1 γ1 ii) C = 0 iii) D = [T] π(γ2) π(γ2) Computational Problems You don’t need to prove your answers to the following questions, but you should still show your work. 10) Let γ = (e1, e2, e3) be the standard basis for R3 . Let γ∗ = (f1, f2, f3) be the dual basis of γ. In this problem we shall consider the following question: What happens to the dual basis if we change only a single vector in γ? Let β = (e1, e2, e1 + e3). Clearly β is also a basis for R3 . Let β∗ = (g1, g2, g3) be the dual basis of β. Find a formula for each functional gi and fi . In other words, find an expression for gi a1 a2 a3 in terms of a1, a2, and a3. Note that β differs from γ only in the third vector. Is g1 = f1 and g2 = f2? Explain why or why not.

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关键词一 关键词二 关键词三 关键词四 关键词五 和法规和的发更好的风格和 ECON 615 FINC 612 CSC148 COMP 639 ACCT 203 ACCT 211 PSYC 101 MTH2009 COMP52315 PSYCO 432 statistics MATH4063 11 xxx 1 MA1MSP MAST30028 MATH3888 CS341 EEE30002 ES2F5 MCD4160 CC0933 ECON 1101 MCHM3001 CHEM3118 COMMGMT 2511 COMP41280 STAT 631 24T1 IRE379 SENG365 ECON433 FIT3139 MATH4602 LING5621M FINS5516 ECON3208 MN2177 L2 SM EEET1415 Finance DS4023 COMP3004 7CCSMRTS K1S5B6 Stat230 CSCI4211 CSI2132 ECON30130 MAS6100 ENVS222 COM2107 INFO1112 STAT 4004 FIT2100 comp2022 COMP4691 Physics 307 Econ 659 probability CSCI316 fir29 CSCI203 COMP3160 COMP6714 INFO1113 Physics 7B Physics Photonics S203031 STAT2005 COMP90015 CSCI 2110 CSCC46 Information CSC477 COMP4650 Statistical COMP 346 COMP1017 ELEC340 MATH1007 Programming function STAT 244 CS 134 Java cse13s Math 108A FW20 ECON3389 CSE 130 Science 331 COMP9020 engineering COSC 445 CSE 5523 GAME2012 visualization MATH 235 Math 370 CSC343 COGS 108 EE524 COMP10200 STAT 416 3FK4 MSCI 311 ECON6113 ACC40700 EMESTER1 MAS8383 ECO 512 STAT 8310 CMN3102 MA577 CMPT 225 COMP0045 Methods Equations MATH 113 Math 3283W ICS-33 EN4302/ENT603 BIA B350F MA318 MTHM506 MECH 5315M ECONOMICS MAS8382 COM 2004 PX3142 XJEL 3662 STAT 231 MGT B399F PX4131 COM2108 30AM MSCI521 ECONM1015 ACFI304 GR5241 MATH10101 ECON 2070 COMS31900 EFIM20011 CSE 250B ACSE-5 STAT4207 SOFT2412 ELEC 331 AY2020 MSIN0209 FIT2094 CS 241L ST404 CS 411 3D CA3 INF4002 COMP3506 SIT718 CS 188 CS 170 MATH 138 CSE 251A CMT118 DATA5207 CMPT307 CMPT459 T1 PGEE11108 COSC 3P32 ME 163 MATH96007 7QQMM203 EE6227 BA 574 STA 4234 ECO206Y1Y ECO2008 MT2300 MATH 11158 CS 527 MACM 316 CSE474 2DI70 ISE 525 COMP3411 EMAT30007 7CCSMBDT MFIN704 MFIN704W21 MfIB ECE 438 CS 436 F36 CS5344 FE 621A FE621 TBA 612 ECM3151 ECMM422 MF1 MSIN0107 ESSMENT CS 5854 MFIT 5008 MIPS R4000 ENG5274 BU52018 ENG2079 BIOA02 STA2570 F0 INF553 STAT 440 STAT3008 COMP9417 COMP6451 6COM1042 BU52006 ST303 HW-1 CISC-235 CS 510 MATH3161 ECN 140 ECE391 Comp 3613 CVEN 9625 PRG455 UESTION 1 EARN 5 FE 825 IERG 5130 STAT 8178 EE161 ADS2 G54FUZ CS326 113B ECN 226 QBUS6860 ELEC3202 COMP9334 MGT11B MET CS 777 STA238 00099F Python SMM306 CMT206 EC116 SP MAST90084 CS480/680 BCPM0091 CMPT 300 ECON2209 FY20 STAT 432 CVEN9625 MATH1712 CMP9771M CMP3750M CIS 325 AI506 MAT344H1S MGT7180 PSTAT 160B CSYS5010 1D STAT 453/558 ME152B STA304H1F CSCI-128 ME428 CO3002 FINE 7640 COMP3234 ECSE 223 EE497 – 3D MATH5905 ECS648U ENG5009 MATH10242 EC665 A CPI221 ISOM5220 MACE40402 MGT B456F COMP9315 AFM 113 STAT 778 CS 564 MA231 CS 230 CS 124 COMP5511 CSC317 ITEC1010 ELEC207 INFT 3019 7CCMFM06 EE 161 5G ECLT5810 MATH 7349 CS526 O2 COMP90054 ELE00063M ACCT5034 ARC1015 CSCI 40 CSCI E40 FIN 524B CS 462 389COM CMT309 MFIN703 COMP4250 STA304 STA304H1S STA 1003 FIN 448 BUFN 732 CS520 CVEN3701 ECMT6006 PHAS0100 CSIT314 MDIA 5028 CSCI 2134 MAT 451 CMPT 459 STAT 457 INFO20003 COMP4141 IELE8066 F540 ECM9 CSE 6321 FIN 430 MATH3911 BFC5916 COMP1011 EEE 6209 STATS 240 CW3 COMP206 COMP1003 4331W ISE 530 CMPT 295 MATH 239 QF5203 INFR08010 IT114105 FIT2093 ECON3124 ECON1195 MATH 337 STAT 3006 COMP202 COMP3130 MATH3609 COMP 250 CS6140 IEEE ELE00041I FIN40090 CIV 5899 MATH3066 IGNMENT MATH 2009 STAT3405 HIST 101 MATH 3281 CEGE0042 ICT532 LAB 7 LAB 5 MCO455 CSIT214 Q4 IFB104 A1 ECO220Y5Y DA5020 CMPSC473 FIN9007 DSCI553 MSCI534 COMP3308 CSE-112 CMPT 310 EE104 F 2C P20 ALY2010 BUSM060 STA258 ECO00030M R BIOSTAT 274 COS6012-B FIT3155 CMPUT 340 EC421 INFS4205 ECOM135 NATS 1740 COMP809 MBE 6005 STA465 CSSE2310 CIVL2700 DATA4700 CMPSC 473 ISE 529 CS5044 MAST30020 COMP2207 ENGG1001 COMP0046 FIT3165 STAT GR5241 MECH0007 vATHK1001 ANALYTIC ATHK1001 MATH352001 HW3 S1 BMAN 24662 COMP 636 Stat-461 STAT 302 STAT 2132 STAR534 ENG4137 CMP3752M CS3103 BISM3222 STA 135 ME155A IIMT3601 ECON3371 ECON 3371 HW4 FINA 5840 DT211C CTEC3902 MAT005 MATH2148 S261F Econ 332 MATH3734 MIS41270 7CCMFM13 SYST 664 CIS434 CENG10014 MT4111 MT4527 ECMM424 CMT219 Math 136 MATH 3120 COMP10002 MAST20009 MATH0058 BEEM117 ISYE 6420 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