Class Note for MATH 290 with Professor Mandal at KU 2
Class Note for MATH 290 with Professor Mandal at KU 2
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Chapter 5 Inner pro duct spaces 51 Length and Dot product in R Homework Textbook7 51 EX 97 117 157 197 237 277 317 397 417 597 677 757 777 857 997 103 p 290 In this section we discuss 1 Length of vectors in R 2 Dot product of vectors in R 3 Cauchy Swartz Inequality in R 4 Triangular Inequality in R 169 170 CHAPTER 5 INNER PRODUCT SPACES De nition 511 We give the main de nitions in this section as fol lows Let u u1u2un V 11111211n be two vectors in R 1 The length or magnitude of vector V is de ned as Hv llv v3v3 a The length of V is also called the norm of V b Also7 if V H 1 then we say V is a unit vector c The de nition shows that V H2 0 and H V H 0 if and only if V 0 2 The distance between u and V is de ned as duV H 11V H xm in U2 12 un v 3 The dot product of u and V is de ned as u VU111u212un1n 4 The angle 6 between u and V is de ned by the formula cos0L 030371 Hullllvll Remark For this de nition to make sense7 we need to assert that 1 1 ll 11 W V H We will prove this later 51 LENGTH AND DOT PRODUCT IN RN 171 Remarks Here are some obvious comments 1 The standard basis vectors ei E R are unit vectors 2 For a nonzero vector V and a nozero scalar CV and CV point to opposite directions Theorem 512 Suppose V 0102 on E R is a vector and C is a scalar Then ll CV H M H V H where lCl denotes the absolute value of C Proof We have CV C111Co2 Ciln Therefore CV H emu W2 0 02 v H3 713 M H v H The proof is complete I Theorem 513 Suppose V 0102 on E R is a nonzero vector Then i V H V H has lenght 1 We say u is the unit vector in the direction of V ProofF239rst note that the statement of the theorem would not make sense unless V is nonzero Now 1 1 H 11 Ml 7V ll 7 H V ll1 H V H H V H The proof is complete I Reading assignment Read Textbook Example 12 p 279 172 CHAPTER 5 INNER PRODUCT SPACES 511 On Distance The distance between uV E R was de ned in the main de nition 5112 as duaV ll 11 V H We have the following proposition Proposition 514 Suppose uV E R are two vectors Then 1 duv 2 0 2 duv dvu 3 duv 0 if and only if u V Proof The proofs follow directly from the de nition of distance I will only prove the last statement We have duv0ltgtHu7VHltgtuiv0ltgtuv The proof is complete I 512 On Dot product The following theorem describes some of the properties of dot product Theorem 515 Suppose uVW E R are three vectors and c is a scalar Then 1 Commutatim39ty 2 Distributimty uVWuvuw 51 LENGTH AND DOT PRODUCT IN RN 173 3 Associatim39ty CuV cu V u CV 4 dot product and Norm v v H v HZ 5 We haveVVZOand V V ltgt V 0 Proof Follows easily from the de nition 511 De nition 516 The vector space R together with 1 length7 2 dot product is called the Euclidian nispace Reading assignment Read Textbook7 Example 3 67 p 282 513 Two Inequalities Theorem 517 CauchySchwartz Inequality Suppose u V E R are two vectors Then7 lu39Vl S llullllVll Proof Case 1 Assume u 0 So7 lUVllOVl0 andllullllvll0HVll0 So7 the inequality is valid if u 0 174 CHAPTER 5 INNER PRODUCT SPACES Case Assume u 31 0 So7 a u u H u H2gt 0 Let t be any real number We have tu V tu V H tu V H22 0 Expanding it7 we have t2uu2tuvVV 2 0 We have a u u H u H2gt 0 and write b 2uV abd c V V So7 the polynomial ft at2 bt c 2 0 for all t So7 ft either has no real root or has a single repeated root By the Quadratic forrnula7 we have b2 7 4ac S 0 07quot b2 S 4ac This means 4UV2 S 4UuVV 4 H 11 W V H2 Taking square root7 we have lUVl S HullllVH The proof is complete I Theorem 518 Triangule Inequality Suppose u V E R are two vectors Then7 HuVH S HullllVll Proof We have H uV H2 uVuV uu2uVVV H u H22UV H V HZSH 11 WW l UV l H V H2 By Cauchy Schwartz lnequality 517 l uV l S u V So7 we get 2 HuV HZSllullZ2 HullllVH llVll2lluH HVH The theorem is established by taking square root I 51 LENGTH AND DOT PRODUCT IN RN 175 De nition 519 Suppose uV E R are two vectors We say that they are orthogonal7 if u V 0 Theorem 5110 Pythagorian Suppose u V E R are two orthogoanl vectors Then H uV H2H u H2 H V H2 Proof From the proof or triangular inequality 518 H uV HZUu2UVVVHu H2 H V H2 The proof is complete I Reading assignment Read Textbook7 Example 7 107 p 285 Exercise 5111 Ex 10 p 290 Let u172717 V07277239 1 Compute u Solution We have H u H 12 22 12 x 2 Compute V Solution We have H v H x02 22 72 V8 3 Compute u V Solution We have H uV HH uV HH 17471 H V12 22 1 V5 176 CHAPTER 5 INNER PRODUCT SPACES Exercise 5112 Ex 16 p 290 Let u 71 34 1 Compute the unit vector in the direction of u Solution First7 H u H 42 32 42 26 The unit vector in the direction of u is e u 77134gt7V1276 32767Q39 HuHi 26 2 Compute the unit vector in the direction opposite of u Solution Answer is l 3 4 e W m Exercise 5113 Ex 24 p 290 Let V be a vector in the same direction as u 7121 and V H 4 Compute V Solution We have V cu with c gt 0 So7 4 HVHHCUHC HuHCW12 22 12 Ch 7 4 7 7 4 Sincecgt0 we haveci 6 andvicuiil21 3 Exercise 5114 Ex 28 p 290 Let V 71304 51 LENGTH AND DOT PRODUCT IN RN 177 1 Find u such that u has same direction as V and one half its length Solution In general7 H CV H M H V H SO7 lIl case 111V11304 1302 2 2 2 2 39 2 Find u such that u has opposite direction as V and one fourth its length Solution Since it has opposite direction 1 1 1 3 u WV 41304 451041 4 4 4 4 3 Find u such that u has opposite direction as V and twice its length Solution Since it has opposite direction u 72V i2 71304 2 76078 Exercise 5115 Ex 32 p 290 Find the distance between u120 and V141 Solution Distance dUV H u i V HH 2 2 1 H 22 2 1 3 Exercise 5116 Ex 40 p 290 Let u 04344 and V 68 733 75 178 CHAPTER 5 INNER PRODUCT SPACES 1 Find u V Solution We have U V 04344 6537319 75 064837343475 15 2 Compute u u Solution We have uu 04344 04344 01691616 57 3 Compute u H2 Solution From 27 we have H u H2uu57 4 Compute u VV Solution From 17 we have u vv 15v 1504344 0 6045 60 60 Exercise 5117 Ex 42 p 290 Let uV be two vectors in R It is given7 Find 3u 7 V u 7 3V Solution We have 3u7Vu73v 3uu710uV3VV 38710736 728 51 LENGTH AND DOT PRODUCT IN RN 179 Exercise 5118 Ex 62 p 291 Let u1710 and V0171 Verify Cauchy Schwartz inequality see 517 Solution We have Hull 121202 and HVH 02121 39 Also UV1071107171 Therefore it is veri ed that lUVl1S2HuHHVH Exercise 5119 Ex 68 p 291 Let u 231 and V 73 2 0 Find the angle 6 between them Solution The angle 6 between u and V is de ned see 511 by the formula cos6 L 0 S 6 S 7139 H 11 M V H Wehave Hull 223212v14 HVH 322202 E and uV27332100 So cos6L0 and 67r2 ll 11 W V H 180 CHAPTER 5 INNER PRODUCT SPACES Exercise 5120 Ex 78 p 291 Let u 2 711 Find all vec tors that are orthogonal to u Solution Suppose V zyz be orthogonal to u By de nition7 it means7 uV2z7yz0 A parametric solution to this system is tysz572t So7 the set of vectors orthogonal to u is given by V tssi2t ts E R Exercise 5121 Ex 82 p 291 Let 1 2 4 3 7 if u lt gt v 2 3 Determine if are u V orthogonal to each other or not Solution We need to check7 if u V 0 or not We have 1 2 4 7 3 if 0 uV 2 3 So7 u V are orthogonal to each other Exercise 5122 Ex 86 p 291 Let u 016 V 1 7271 Determine if are u V orthogonal to each other or not Solution We need to check7 if u V 0 or not We have uV0112671777 0 So7 u V are not orthogonal to each other 51 LENGTH AND DOT PRODUCT IN RN 181 Exercise 5123 Ex 100 p 292 Let u 111 v 0172 Verify triangle Inequality see 518 Solution We have HuHV121212 Hv H v021272 VS and H uV HH 17271 H V1222 1 5 We need to Check HuV H26 S H u H2 HV H215 So the triangle inequality is veri ed Exercise 5124 Ex 104 p 292 Let u 3 72 V 46 Verify Pythagorian Theorem see 5110 Solution We have u V 3 gtk 4 7 2 gtk 6 0 So uV are orthogonal to each other and Pythagorian Theorem see 5110 must hold l u H 32lte2gt V H v H W and H u V 74 H V72 42 V65 We need to Check ll uV H2 65 ll 11 H2 H V H21352 So the Pythagorian Theorem is veri ed 182 CHAPTER 5 INNER PRODUCT SPACES 52 Inner product spaces Homework Textbook7 EX 37 57 77 117 137 157 597 61 p 303 In this section we de ne abstract inner product spaces The concepts of length and dot product on the Euclidean spaces R is emtended to vector spaces with lnnner products as follows De nition 521 Suppose V is a vector space An inner product on V is a function lt gtk gt V gtlt V a R that associates each pair u V of elements in V to a real number lt uV gt such that for all uVW in V and scalar c we have 1 ltuVgtltVugt 2 ltuVWgtltuVgtltuWgt 3 cltuVgtlt cuVgt 4 ltVVgt20andp0ltgtltvvgt0 A vector space V together with an inner product lt gtk gt is called an inner product space For such an inner product space7 1 The length of a vector V E V is de ned as H V H xlt VV gt The length V H is also called the norm of 1 52 INNER PRODUCT SPACES 183 2 The distance between two vectors u V E V is de ned as dbl V ll 11 7 V H 3 The angle 6 between two vectors uV E V is de ned by the for rnula lt uV gt cos0 Hullllvll 030371 Example 522 For uv E R de ne lt uV gt u V This is an inner product on R So7 R together with dot product is an inner product space A better and nontrivial example is Textbook7 Example 57 which discuss as follows Example 523 Let V Cab be the vector space of all continuous functions f ab a R For fg E Cab de ne inner product 17 lt fag gt fzgzdz It is easy to check that lt fg gt satis es the properties of de nition 521 of inner product space Narnely7 we have 1 lt fg gtlt gfgt for all fg E Chub 2 ltfghgtlt fggtltfh gtfor all fgh Cabl 3 cltfg gtlt cfg gt for all fg Cab anchR 4 ltffgt20forallf 0abandf0 gtltffgt0 184 CHAPTER 5 INNER PRODUCT SPACES Accordingly7 for f E Cab we can de ne length or norm 7 VilaW mm This length will have all the properties that you mpect length to have The following are some properties of inner product Theorem 524 Let V be an inner product space and u V W E V and c E R be a scalar Then7 1 lt V0 gtlt 0u gt 0 2 ltuvwgtltuwgtltvwgt 3 ltucv gtcltuvgt Proof All these three staternents follows from curnrnutatiVity7 proerty 1 of de nition 521 First7 the rst equaity of 1 follows from curnrnutatiVity7 proerty 1 of de nition 521 Then7 we have ltv0gtltv00gtltv0gtltv0gt Now7 subtracting lt V0 gt from both sides7 we get lt V0 gt 0 To prove 27 we have lt uvw gtlt Wuv gtlt Wu gt lt WV gtlt uw gt lt VW gt To prove 37 we have ltucv gtlt CVVgtC lt Vu gtc ltuv gt The proof is complete I Theorem 525 Let V be an inner product space and u V E V Then7 52 INNER PRODUCT SPACES 185 1 Cauchy Schwartz Inequality lltmVgtlSWWHVW 2 Triangle Inequality HuVHSHuHHVH 3 De nition We say uV are mutually orthogonal if lt uv gt 0 In this case we write u 1 V and say they are perpendicular to each other 4 If uV are orthogonal then HuVWWHVHVW This is called Pythagorean Theorem Proofs The proofs are exactly line for line similar to that of the corresponding theorerns in section 51 1 To prove 1 Cauchy Schwartz Inequality repeat the proof of the orern 517 2 To prove 2 the Triangle Inequality repeat the proof of theorem 518 3 To prove the Pythagorean Theorern repeat the proof of theorem 5110 So the proofs are complete I 186 CHAPTER 5 INNER PRODUCT SPACES 521 Orthogonal Projections De nition 526 Let V be an inner product space Suppose V E V is a vector Then7 lt gt for u E V define projvu v It is easy to check that u 7 projvu l projvu Reading assignment Read Textbook7 Example 1 87 p 293 Exercise 527 Ex 4 p303 ln R2 de ne an inner product for u u1u2 V 01112 define lt uV gt mm 2u2112 It is easy to check that it de nes an inner product as de ned in 521 Now let u0776a V711139 1 Compute lt uV gt Solution We have lt uV gt mm 2u202 0 gtk 71 276 gtk l 712 2 Compute u Solution We have H u H lt uu gt xulul 2u2u2 xO gtk 0 276 gtk 76 v72 3 Compute V Solution We have H v H W even 21 lt1 3 52 INNER PRODUCT SPACES 187 4 Compute du V Solution We have dUV H u i V HH 1 7 H 11 2777 x Exercise 528 Ex 12 p 303 Let V C711 with inner prod uct 1 lt 129 gt1f96996d96 for 1296 V Let fx 7 and g x2 7 z 2 1 Compute lt fg gt Solution We have 1 lt fag gt 11fltzgtgltzgtdz 71 il2l ii21 3 7 4 3 2 4 3 2 339 2 Compute norm f Solution We have x W 3 Compute norm g Solution We have H 9 km 19z2dz 1z2x22dz 188 CHAPTER 5 INNER PRODUCT SPACES 1 x47235x274z4dx 2108 5 3 39 x5 4 3 2 1 7727 57747 4 5 4 3 2 471 4 Compute dfg Solution We have df9 H f 7 g H lt ff gt 7 MAezz 7 am 1 5 3 z44z24dx 7474z 1 5 3 1 2 8 17 7 8 53L Exercise 529 Ex 60 p 305 Let V C711 with inner prod uct 1 lt 129 gt1f9c996d96 for 1296 V Let fx z and g Show that f and g are orthogonal Solution We have to show that lt fg gt 0 We have 1 1 1 lt fg gt fzgzdz z32271dz 3z3 7 zdz 1 7 71 1 4 2 1 1 319 1 3L1 l 31 2 4 2 71 2 4 2 2 4 2 Song IAS Slavic amp Eurasian 2008 Fall Eu asian Security and Geopolitics Giullian 1 Slavic amp Eurasian Studies Dept 519 Watson Library Level 5 EVRN 420 GEOG 571 POLS 689 REES 480 Library Research Session ii Sep2008100215AIC Jon Giullian 8648854 Geoff Husic 8643957 giulliankuedu husickuedu TR Smith Maps Collection Scott McEathron 8644662 macmapSGQkuedu News sources Research by Subject gt Area Studies gt REES gt Articles amp Databases Databases The Current Of the POSt Soviet Press Provides translations or abstracts of materials selected from awide variety of Russianelanguage newspapers and periodicals Covers political reform public health privatization foreign policy and international affairs and other social cultural and legal issues East View Universal Databases Central Newspapers UDBCOM iThe Central Newspapers module contains the fulletext of over 50 of the most in uential Russian and NlS periodicals on the same day they are published The archives of back issues are a source for information on business economics domestic and foreign policy and important political events in Russia and the former Soviet Union East View Universal Databases CIS and Baltic Publications U DBCIS 7 Fulletext articles from authoritative newspapers and periodicals from Central Asia Caucasus and Baltics Sources are in Russian and English covering issues ofdomestic and international importance LexisNexis Academic Provides fulletext articles from domestic and foreign news sources 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sciences Topics covered include arts business children education general interest health humanities international studies law military multicultural studies psychology sciences social sciences and women39s interests Web Of Science Social Sciences Citation Index 7 a multidisciplinary index to the journal literature of the social sciences It fully indexes more than 1725 journals across 50 social sciences disciplines and it indexes individually selected relevant items from over 3300 of the world39s leading scientific and technical journals Wilson Om nifile Provides indexing abstracting and full7text content to journal articles in a variety of disciplines Jon Giullian Sm floor Watson Library 8648854 giulliankuedu Geoff Husic Sm floor Watson Library 8643957 husickuedu Page 2 of 4 IAS Slavic amp Eurasian 2008 Fall Databases Russian content Giullian 3 Worlclwicle Political Science Abstracts iAbstracts and indexes journals in political science international relations law public administration 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and countries in Eastern Europe Original script display is available for journals published in Cyrillic Websites on Central Asia security geopolitics environment etc Government and Military Contacts htt wwweiadoe ov emeu cabs Cas ian Back roundhtml ISN International Relations and Security Network http11wwwisnethzch1 Jupiter Jamestown Foundation http11wwwjamestownorgn Center for Strategic and International Studies httpwwwcsisorg The Brookings Institution http11wwwbrookingsedu1 Central Asia Caucasus Institute Silk Road Studies Program httDwwwsilkroadstudiesorqnewindexhtm George C Marshall European Center for Security Studies MERLN Military Education Research Library Network httpmerln ndued uindexcfmsecID 153ampDaquD3amptVDesection Erik Herron s Guide to Post Communist States on the Web httpwebkueduzherron Jupiter Central Asia Studies Worldwide httDCeswwfasharvardeduindexhtml Jupiter Info Gateway gt Area Studies gt Databases gt REES Jon Giullian Sm floor Watson Library 8648854 giulliankuedu Geoff Husic Sm floor Watson Library 8643957 husickuedu Page 3 of 4 IAS Slavic amp Eurasian 2008 Fall Giullian 4 East View Universal Databases Government Publication UDBGOV e Monitors mainly the events in the Federal Assembly of the Russian Federation Provides access to Russian government documents stenographic records of Federation Council and Burma hearings draft legislation official resolutions vote results and other papers of record East View Universal Databases Military Publications UDBCOM 7Provides access to a broad range of military texts including central military journals and regional military district newspapers in all major strategic areas from both official and independent sources ON WWW Yahoo Directory httpdiryahoocomgovernmentcountiies 7 Has substantial information for every nation state in Eastern Europe and Eurasia Categories include Conventions and Conference Documents Embassies and Consulates Ethics Government Officials Intelligence Judicial Branch Law Legislative Branch lVIilitary lVIinistizies News and Media Politics Erik Herron39s Guide to PostCommunist States on the Web httpwwwkueduNherron 7 An award winning database of 3000 links to sites about politics and economics in the postecommunist states of East Central Europe and Eurasia OQJMUMaanaFI POCCVIFI http1 wwwggvru 7 Russian Federation s directory to government information MMHMCTepCTBO o6opom Russian Ministry of Defense httpwwwmilru 7 Includes a biographical dictionary of military figures BoeHHajr IICTOPIISI B Anuax and the soldier s calendar KaAeHAapb BerIna Portals on the Web Contacts Yandex httpwwwyandexru The Russian equivalent onahoo Very powerful upetoedate search engine Ramblerru httpwwwramblerru 7 One of the oldest and most popular Russian portals similar to Yahoo Listru httplistmailruindexhtml 7 Popular Russian directory similar to the Google directory interface Jon Giullian Sm floor Watson Library 8648854 giulliankuedu Geoff Husic Sm floor Watson Library 8643957 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