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# Modern Linear Algebra MAT 067

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This 16 page Class Notes was uploaded by Otilia Murray I on Tuesday September 8, 2015. The Class Notes belongs to MAT 067 at University of California - Davis taught by Anne Schilling in Fall. Since its upload, it has received 59 views. For similar materials see /class/187336/mat-067-university-of-california-davis in Mathematics (M) at University of California - Davis.

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MAT067 University of California Davis Winter 2007 Linear Span and Bases Isaiah Lankham Bruno Nachtergaele Anne Schilling January 23 2007 Intuition probably tells you that the plane R2 is of dimension two and the space we live in R3 is of dimension three You have probably also learned in physics that space time has dimension four and that string theories are models that can live in ten dimensions In these lectures we will give a mathematical de nition of what the dimension of a vector space is For this we will rst need the notion of linear spans linear independence and the basis of a vector space 1 Linear span As before let V denote a vector space over F Given vectors 11112 1 E V a vector 1 E V is a linear combination of 111om if there exist scalars a1 am 6 IF such that 1 a111 12112 amiim De nition 1 The linear span or simply span of 111 11 is de ned as spano1om a1o1 amom i a1 am 6 Lemma 1 Let V be a vector space and 111112 1 E V 11 E spaho1o2 om 2 spaho1o2 11 is a subspace of V 3 IfU C V is a subspace such that 111112 1 E U theh spaho1o2 11 C U Proof 1 is obvious For 2 note that 0 E spano1o2 11 and that spano1o2 11 is closed under addition and scalar multiplication For 3 note that a subspace U of a vector space V is closed under addition and scalar multiplication Hence if 111om E U then any linear combination alol amom must also be in U Copyright 2007 by the authors These lecture notes may be reproduced in their entirety for non commercial purposes 2 LINEAR INDEPENDENCE 2 Lemma 1 implies that span13171327 71 is the smallest subspace of V containing all U17U2771m De nition 2 If span1317 713m V7 we say that 1317 713m spans V The vector space V is called nitedimensional7 if it is spanned by a nite list of vectors A vector space V that is not nite dirnensional is called in nitedimensional Example 1 The vectors 61 107707 62 017 707en 077071 span F Hence F is nite dirnensional Example 2 lf132 am2m am12m 1 112 a0 6 73lF is a polynomial with coef cients in l such that am 31 0 we say that 132 has degree m By convention the degree of 132 0 is foo The degree of 132 is denoted by deg132 De ne 73mlF set of all polynomials in 73lF of degree at most m Then 73mlF C 73lF is a subspace since it contains the zero polynomial and is closed under addition and scalar rnultiplication In fact 73mlF span127227 72 quot Example 3 We showed that 73lF is a vector space In fact7 73lF is in nite dirnensional To see this7 assume the contrary7 namely that 73lF span13127 713k2 for a nite set of k polynornials 13127 713k2 Let m rnaxdeg13127 7deg13 2 Then 2m1 E 73G7 but 2quot 1 span13127 713k2 2 Linear independence We are now going to de ne the notion of linear independence of a list of vectors This concept will be extremely important in the following7 especially when we introduce bases and the dimension of a vector space De nition 3 A list of vectors 1317 71 is called linearly independent if the only solution for 117 7am E l to the equation a1 1amvm0 is al am 0 In other word the zero vector can only be trivially written as the linear combination of 1317 713m 2 LINEAR INDEPENDENCE 3 De nition 4 A list of vectors o17 mm is called linearly dependent if it is not linearly independent That is7 there exist ah am 6 l9 not all being zero such that a1o1amom0 Example 4 The vectors e177em of Example 1 are linearly independent The only solution to 0a161am6m 117quot397am is a1am0 Example 5 The vectors 17 27 72m in the vector space 73mlF are linearly independent Requiring that a01a12amzm0 means that the polynomial on the left should be zero for all z E F This is only possible for a0a1am0 An important consequence of the notion of linear independence is the fact that any vector in the span of a given list of linearly independent vectors can be uniquely written as a linear combination Lemma 2 The list of vectors 11771m is linearly independent if and only if every 1 E spano17 mm can be uniquely written as a linear combination of 1117 7om Proof 77gt Assume that 1117 7om is a linearly independent list of vectors Suppose there are two ways of writing 1 E spano17 7om as a linear combination of the vi 1 a101 39 39 39am m i aaol anom Subtracting the two equations yields 0 a1 7 a ol am 7 anom Since 1117 mm are linearly independent the only solution to this equation is al 7 a3 07 am 7 an 07 or equivalently a1 a3 7am an 77lt Now assume that for every 1 E spano17 7om there are unique ah am 6 l9 such that oa1o1amom This implies in particular that the only way the zero vector i 0 can be written as a linear combination of 01771m is with al am 0 This shows that 11771m are linearly independent 2 LINEAR INDEPENDENCE 4 It is clear that if 11771m is a list of linearly independent vectors then the list 1117 7om1 is also linearly independent For the next lemma we introduce the following notation If we want to drop a vec tor oj from a given list 11771m of vectors7 we indicate the dropped vector by a hat 117717j71m Lemma 3 Linear Dependence Lemma If 117 7om is linearly dependent and U1 31 0 there edists an indecoj E 27 7771 such that 1 vi 6 spano177oj1 2 fr is removed from o17 mm then spano17 71777 7om spano17 7om Proof Since 11771m is linearly dependent there exist 117 7am E l not all zero such that 11111 amom 0 Since by assumption o1 31 07 not all of 127 7am can be zero Let j E 27 7771 be largest such that 17 31 0 Then we have 11 173971 o77o77 11 1 J aj 1 aj J 17 which implies part 1 Let 1 E spano17 mm By de nition this means that there exist scalars b17 7bm E l such that ob1o1bmom The vector oj that we determined in part 1 can be replaced by 17 so that o is written as a linear combination of o17 71377 mm Hence spano17 71377 7om spano17 7om 7 17 27 170 of vectors in R2 They span R2 We want to show that 1 can be written as a Example 6 Take the list 01702703 To see this7 take any vector 1 Ly 6 linear combination of 1717 1727170 171 R2 vm0wdmdm or equivalently w a1 a2 a37a1 2a2 Taking a1 y7a2 0713 z 7 y is a solution for given Ly E R Hence indeed R2 span17 17 17 27 17 Note that 217171727170 0707 2 which shows that the list 17 17 17 27 17 0 is linearly dependent The Linear Dependence Lemma 3 states that one of the vectors can be dropped from 17171727170 and still 3 BASES 5 span R2 Indeed by 2 U3 170 2171 i 172 2m 7 so that span17 17 17 27 17 0 span17 17 17 The next results shows that linearly independent lists of vectors that span a nite dimensional vector space are the smallest possible spanning sets Theorem 4 Let V be a nite dimensional vector space Suppose that 11771m is a linearly independent list of vectors that spans V and let wL7 7ion be any list that spans V Then in S n Proof The proof uses an iterative procedure We start with an arbitrary list SO wL7 7ion that spans V At the h th step of the procedure we construct a new list 8k by replacing a wjk by ilk such that 8k still spans V Repeating this for all ilk nally produces a new list Sm of length n that contains all 1117 7pm This proves that indeed in S ii Let us now discuss each step in the procedure in detail Step 1 Since w177wn spans V7 adding a new vector to the list makes the new list linearly dependent Hence 111711117 710 is linearly dependent By Lemma 3 there exists an index jl such that wjl E spano17w17 qua11 Hence 81 o17w177ifj177wn spans V In this step we added the vector U1 and removed the vector wjl from 80 Step k Suppose that we already added 1117 7ok1 to our spanning list and removed the vectors wj177wjk71 in return Call this list Sk1 which spans V Add the vector ilk to Sk1 By the same arguments as before7 adjoining the extra vector ilk to the spanning list Sk1 yields a list of linearly dependent vectors Hence by Lemma 3 there exists an index jk such that Sk1 with ilk added and wjk removed still spans V The fact that 11771k is linearly independent ensures that the vector removed is indeed among the 10 Call the new list 8k which spans V The nal list Sm is SO with all 1117 711m added and 112717 7mm removed It has length n and still spans V Hence necessarily in S n D 3 Bases A basis of a nite dimensional vector space is a spanning list that is also linearly independent We will see that all bases of nite dimensional vector spaces have the same length This length will be the dimension of our vector space 3 BASES 6 De nition 5 A basis of a nite dimensional vector space V is a list of vectors 11 07 in V that is linearly independent and spans V lf 11 1m forms a basis of V then by Lemma 2 every vector 1 E V can be uniquely written as a linear combination of 11 1m Example 7 e1 en is a basis of F There are of course other bases For example 12 1 1 is a basis of F2 The list 11 is linearly independent but does not span F2 and hence is not a basis Example 8 1 z 22 27quot is a basis of Theorem 5 Basis Reduction Theorem IfV span11 vm then some 1 etm be removed to obtain a basis of V Proof Suppose V span11 1m We start with the list 8 11 1m and iteratively run through all vectors 1 for k 12 m to determine whether to keep or remove them from 8 Step 1 H111 0 remove 01 from 8 Otherwise leave 8 unchanged Step k lf 1 E span11 vk1 remove 11k from 8 Otherwise leave 8 unchanged The nal list 8 still spans V since at each step a vector was only discarded if it was already in the span of the previous vectors The process also ensures that no vector is in the span of the previous vectors Hence by the Linear Dependence Lemma 3 the nal list 8 is linearly independent Hence 8 is a basis of V Example 9 To see how Basis Reduction Theorem 5 works consider the list of vectors 8 717 07 27 727 07 717 07 17 07 717 17 07 17 This list does not form a basis for R3 as it is not linearly independent However it is clear that R3 spanS since any arbitrary vector 1 zyz E R3 can be written as the following linear combination over S 1 m 21 710 02720 z710 1 00 711 m y z0 10 In fact since the coef cients of 2 720 and 0711 in this linear combination are both zero it suggests that they add nothing to the span of the subset B 1 710 710 1 0 1 0 of 8 Moreover one can show that B is a basis for R3 and it is exactly the basis produced by applying the process from the proof of Theorem 5 as you should be able to verify 4 DIMENSION 7 Corollary 6 Every nite dimensional vector space has a basis Proof By de nition a nite dirnensional vector space has a spanning list By the Basis Reduction Theorem 5 any spanning list can be reduced to a basis Theorem 7 Basis Extension Theorem Every linearly independent list of vectors in a nite dimensional vector space V can be emtended to a basis of V Proof Suppose V is nite dirnensional and 111 vm is linearly independent Since V is nite dirnensional there exists a list wl wn of vectors that spans V We wish to adjoin some of the wk to v1 11 to create a basis of V Step 1 If wl E spanv1 vm let S v1 vm Otherwise set S 111 vmw1 Step k If wk 6 spanS leave 8 unchanged Otherwise adjoin wk to 8 After each step the list 8 is still linearly independent since we only adjoined wk if wk was not in the span of the previous vectors After n steps wk 6 spanS for all h 12 n Since wl wn was a spanning list 8 spans V so that S is indeed a basis of V D 4 Dimension We now come to the important de nition of the dimension of nite dirnensional vector spaces lntuitively we know that the plane R2 has dimension 2 R3 has dimension 3 or more generally R has dimension n This is precisely the length of the bases of these vector spaces which prompts the following de nition De nition 6 We call the length of any basis of V which is well de ned by Theorem 8 the dimension of V also denoted dirn V Note that De nition 6 only makes sense if in fact all bases of a given nite dirnensional vector space have the same length This is true by the next Theorern Theorem 8 Let V be a nite dirnensional vector space Then any two bases ofV have the same length Proof Let 111 vm and w1 wn be two bases of V Both span V By Theorem 4 we have m S n since v1 11 is linearly independent By the same theorem we also have n S in since w1 wn is linearly independent Hence n m as asserted Example 10 dirn IF n and diumGF m 1 Note that dirn C n as a C vector space but dirn C 2n as an R vector space This comes from the fact that we can view C itself as an R vector space of dimension 2 with basis Li Theorem 9 Let V be a nite dimensional vector space with dirn V n Then 4 DIMENSION 8 Z IfU C V is a subspace of V then dimU S dim V 2 UV spani1 71 then m 7on is a basis of V 3 If117 7on is linearly independent in V then 111 71 is a basis of V Point 1 implies in particular7 that every subspace of a nite dimensional vector space is nite dimensional Points 2 and 3 show that if the dimension of a vector space is known to be n7 then to check that a list of n vectors is a basis it is enough to check whether it spans V resp is linearly independent Proof To prove point 17 let uL7 7um be a basis of U This list is linearly independent both in U and V By the Basis Extension Theorem 7 it can be extended to a basis of V7 which is of length n since dim V n This implies that m S n as desired To prove point 2 suppose that 111 71 spans V Then by the Basis Reduction Theo rem 5 this list can be reduced to a basis However7 every basis of V has length n7 hence no vector needs to be removed from 111 7on Hence 117 7on is already a basis of V Point 3 is proved in a very similar fashion Suppose 117 7on is linearly independent By the Basis Extension Theorem 7 this list can be extended to a basis However7 every basis has length n7 hence no vector needs to be added to o17 7on Hence o17 7on is already a basis of V D We conclude this chapter with some additional interesting results on bases and dimen sions The rst one combines concepts of bases and direct sums of vector spaces Theorem 10 Let U C V be a subspace of a nite dirnensional vector space V Then there epists a subspace W C V such that V U 69 W dim V Hence by the Basis Extension Theorem 7 uL7 7um can be extended to a basis uh 7lira117 710 of V Let W spanw1 7wn To show that V U 69 W7 we need to show that V U W and U W Since V spanu1 7um7w17 quot710 and uh 7um spans U and wh 710 spans W7 it is clear that V U W To show that U W 0 let i E U W Then there exist scalars a17am7b177bn 6 IF such that Proof Let ul7um be a basis of U By point 1 of Theorem 9 we know that m lt va1U1amu7nb1w1bnwn or equivalently a1U1amu1n7b1w17ebnwn0 Since uL7 7lira117 710 forms a basis of V and hence is linearly independent7 the only solution to this equation is al am b1 bn 0 Hence 1 07 proving that indeed U W D 4 DIMENSION 9 Theorem 11 If U7 W C V are subspaces of a mte dz39mensz39onal vector space then dimU W dim U dim W 7 dimU Proof Let 1177vn be a basis of U W By the Basis Extension Theorem 77 there exist uh7uk and w177wl such that 117711mu177uk is a basis of U and 1117 7117L401 7wl is a basis of W lt suf ces to show that B 1771mu177uk7w177wzgt is a basis of UW7 since then dimUWnkZnknZindimUdimWidimU W Clearly span117 7vmu17 7uk7w17 7101 contains U and W and hence UW To show that B is a basis it hence remains to show that B is linearly independent Suppose alvl l quotl anvn l blulquot bkukclw1quot clwl0 3 Letu Cali1anvnb1u1bkuk E U Then by 3 alsou iclwliuiclwl 6 W7 which implies that u E U W Hence there exist scalars 137 7a 6 l9 such that u 13111 ally Since there is only a unique linear combination of the linearly independent vectors 117 711mm 7uk that describes u7 we must have b1 bk 0 and a1 1177an all Since 1117 7117L401 7wl is also linearly independent7 it further follows that al an cl cl 0 Hence 3 only has the trivial solution which implies that B is a basis 1 MAT067 University of California Davis Winter 2007 Some Common Mathematical Symbols and Abbreviations with History Isaiah Lankham Bruno Nachtergaele Anne Schilling January 21 2007 Binary Relations the equals sign means is the same as and was rst introduced in the 1557 book The Whetstone of Witte by Robert Recorde c 1510 1558 He wrote I will sette as l doe often in woorke use a paire of parralles or Gemowe lines of one lengthe thus bicause noe 2 thynges can be moare equalle Recorde used an elongated form of the modern equals sign the less than sign mean is strictly less than and gt the greater than sign means is strictly greater than They rst appeared in Artis Analyticae Prams ad Aequationes Algebraicas Resoluendas The Analytical Arts Applied to Solving Algebraic Equations by Thomas Harriot 1560 1621 which was published posthumously in 1631 Pierre Bouguer 1698 1758 later re ned these to less than or equals and 2 greater than or equals in 1734 the equal by de nition sign means is equal by de nition to This is a common alternate form of the symbol Def which appears in the 1894 book Logica Matematica by the logician Cesare Burali Forti 186171931 Other common alternate forms of the symbol Def include dEf and E the latter being especially common in applied mathematics Some Symbols from Mathematical Logic 9 gt ltgt three dots means therefore and rst appeared in print in the 1659 book Teusche Algebra Teach Yourself Algebra by Johann Rahn 1622 1676 the such that sign means under the condition that However it is much more common and less ambiguous to just abbreviate such tha as st the implies sign means logically implies that Eg if it s raining then it s pouring is equivalent to saying it s raining it s pouring The history of this symbol is unclear the iff sign means if and only if and is used to connect logically equivalent statements Eg it s raining iff it s really humid means simultaneously that if it s raining then it s Copyright 2007 by the authors These lecture notes may be reproduced in their entirety for noncommercial purposes really humid and that if it s really humid then it s raining In other words the statement it s raining implies the statement it s really humid and Vice versa This notation iff is attributed to the great mathematician Paul R Halmos 191672006 V the universal quanti er symbol means for all and was rst used in the 1935 publication Untersuchungen ueber das logische Schliessen Investigations on Logical Reasoning by Gerhard Gentzen 1909 1945 He called it the All Zeichen all character in analogy with 3 read there exists NJ the existential quanti er means there exists and was rst used in the 1897 book For mulaire de mathematiqus by Giuseppe Peano 1858 1932 B the Halmos tombstone means QED which is an abbreviation for the Latin phrase quad erat demonstrandum which was to be proven QED has been the most common way to symbolize the end of a logical argument for many centuries but the modern convention in mathematics is to use the tombstone in place of QED This tombstone notation is attributed to the great mathematician Paul R Halmos 19167 2006 Some Notation from Set Theory C the is included in sign means this set is a subset of and D the includes sign means this set has as a subset They were introduced in the 1890 book Vorlesungen iiber die Algebra der Logik Lectures on the Algebra of the Logic by Ernst Schroder 1841 1902 m the is in sign means is an element of and rst appeared in the 1895 book Formulaire de mathematiqus by Giuseppe Peano 1858 1932 Peano originally used the Greek letter 6 which is the rst letter of the Latin word est meaning is but it was Betrand Russell 1872 1970 in his 1903 Principles of Mathematics that introduced the modern stylized version C the union sign means take the elements that are in either set and O the intersection sign means take the elements that the two sets have in common They were introduced in the 1888 book Calcolo geometrico secondo l Ausdehnungslehre di H Grassmarm preceduto dalle operazioni della logica deduttiua Geometric Calculus based upon the teachings of H Grassman preceded by the operations of deductive logic by Giuseppe Peano 1858 1932 0 the null set or empty set symbol means the set without any elements in it and was rst used in the 1939 book Elements de mathematique by N Bourbaki a group of primarily European mathematiciansinot a single person It was borrowed simultaneously from the Norwegian Danish and Faroese alphabets by group member Andre Weil 1906 1998 00 in nity denotes a number of arbitrarily large magnitude and rst appeared in print in the 1655 book De Sectionibus Conicus On Conic Section by John Wallis 1616 1703 Conjectured explanations for why Wallis used this symbol include its resemblance to the symbol 00 used by Romans to denote the number 1000 its resemblance to the nal letter of the Greek alphabet w and so is synonymous with being the nal number and the symbolism of the fact that one can traverse a given curve in nitely often Some Important Numbers in Mathematics 7139 the ratio of the circumference to the diameter of a circle denotes the number 3141592653589 and was rst used by William Jones 1675 1749 in his 1706 book Syn opsis palmariorum mathesios A New Introduction to the Mathematics However it was Leonhard Euler 1707 1783 who rst popularized the use of the letter 7139 for this number in his 1748 book Introductio in Analysin In nitorum Many people speculate that Jones chose the letter 7139 because it s the rst letter in Greek word peiimetron wepipeTpozl which roughly means around 5 limnnoo1 the natural logarithm base denotes the number 2718281828459 and was rst used by Leonhard Euler 1707 1783 in the manuscript Meditatio in Erpeiimenta erplosione tormentorum nuper instituta Meditation on experiments made recently on the ring of cannon written when he was just 21 years old Note that e is the rst letter in exponential The very famous mathematician Edmund Landau 1877 1938 once wrote that The letter 5 may now no longer be used to denote anything other than this positive universal constant 39y limHooZZ1 iiln n the EulerMascheroni constant denotes the number 0577215664901 and was rst used by Lorenzo Mascheroni 1750 1800 in his 1792 Adnotationes ad Euleii Cal culum Integralem Annotations to Euler s lntegral Calculus The number 39y is usually considered to be the third most important important non basic number in mathematics following closely 7139 and e V71 the imaginary unit was rst used by Leonhard Euler 1707 1783 in his 1777 memoir Institutionum calculi integralis Foundations of Integral Calculus s Appendix Some Common Latin Abbreviations and Phrases Cf also http en wikipedia orgwikiListofLatinphrases ie id est means that is or in other words It is used to paraphrase a statement that was just made not to mean for example and is always followed by a comma eg exempli gratia means for example It is usually used to give an example of a statement that was just made and is always followed by a comma Viz uidelicet means namely or more speci cally It is used to clarify a statement that was just made by providing more information and is never followed by a comma etc et cetera means and so forth or and so on It is used to suggest that the reader should infer further examples from a list has been started and is usually not followed by a comma et al et alii means and others It is used in place of listing multiple authors past the rst and is never followed by a comma It s also an abbreviation for et alibi means and elsewhere O H conferre means compare to or see also It is used either to draw a comparison or to refer the reader to somewhere they can nd more information and is never followed by a comma qv quod uide means which see or go look it up if you re interested It is used to cross reference a different work or part of a work and is never followed by a comma The plural form is qq V U uide supra means see above It is used to imply that more information can be found before the current point in a written work and is never followed by a comma NB Nata Bene means note well or pay attention to the following It is used to imply that the wise reader will pay especially careful attention to the what follows and is never followed by a comma vs uersus means against7 or in contrast to It is used to contrast two things and is never followed by a comma c circa means around or near It is used when giving an approximation7 usually for a date7 and is never followed by a comma It s also commonly written as ca 7 cir circ 7or ex lib ex libris means from the library of It is used to indicate ownership of a book and is never followed by a comma o d fortiori means from the stronger or more importantly o a priori means from before the fact and refers to reasoning done before an event happens 0 d posterimi means from after the fact and refers to reasoning done after an event happens ad hoc means to this and refers to reasoning that is quite speci c to an event as it is happening Such reasoning is usually considered to not generalize to other situations very well 0 ad in nitum means to in nity7 or without limit 0 ad nauseam means causing sea sickness or to excessive o mutatis mutandis means changing what needs changing or with the necessary changes 0 non sequitur means it does not follow and refers to something that is out of place in a logical argument This is sometimes abbreviated as non seq 0 Me transmitte sursum Cdledoni means Beam me up7 Scottyl o Quid quid latine dictum sit altum uidetur means Anything said in Latin sounds profound MAT067 University of California Davis Winter 2007 Introduction to Complex Numbers Summary Isaiah Lankham7 Bruno Nachtergaele7 Anne Schilling January 147 2007 1 De nition of Complex Numbers Let R denote the set of real numbers We will denote the set of samplers numbers by C Here is the de nition De nition 11 The set of samplers numbers C is de ned as C 9679 l MI E R For any complex number 2 Ly7 we call Rez a the real part of z and lmz y the imaginary part of 2 In other words7 we are de ning a new collection of numbers 2 by taking every possible ordered pair Ly of real numbers my 6 R7 and z is called the real part of the ordered pair my to imply that the set of real numbers R should be identi ed with the subset 70 i z E R C C It is also common to use the term purely imaginary for any complex number of the form 07y7 where y E R In particular7 the complex number 071 is special7 and it is given the name imaginary unit It is standard to denote it by the single letter i or j ifi is being used for something else7 such as for electric current in Electrical Engineering Note that z Ly x170y071 z1yi We usually write 2 iy It is often signi cantly easier to perform arithmetic operations on complex numbers when written in z iy77 notation7 rather than the ordered pair notation of the de nition 2 Operations on Complex Numbers 21 Addition and Subtraction of Complex Numbers Addition of complex numbers is performed component wise7 meaning that the real and imag inary parts are simply combined Copyright 2006 by the authors These lecture notes may be reproduced in their entirety for non commercial purposes 2 OPERATIONS ON COMPLEX NUMBERS 2 De nition 21 Given two complex numbers phyl x2y2 E C we de ne their complep sum to be 17111 27112 1 27111 12 Example 22 3 2 17 745 3 172 i 45 20 725 As with the real numbers subtraction is de ned as addition of the opposite number aka the additive inverse of z Ly which is de ned as 72 ix 7y Example 23 7T i wax19 7r 7 7T27 M19 7T27 M19 The addition of complex numbers shares a few other properties with the addition of real numbers including associativity commutativity the existence and uniqueness ofthe additive identity or neutral element denoted by 0 and the existence and uniqueness ofthe additive inverse already mentioned above We summarize these properties in Theorem 24 below Theorem 24 Let 212223 6 C be any three complecc numbers Then the following state ments are true 1 Associativity 21 22 23 21 Z2 23 2 Oommutativity 21 22 22 21 3 Additive Identity There is a unique complecc number denoted 0 such that 0 21 21 Moreover 0 00 4 Additive Inverses Given 2 E C there is a unique complecc number denoted 72 such that z 72 0 Moreover ifz zy with my 6 R then 72 ix 7y The proof of this Theorem is straightforward Just use the de nition of when used to denote the addition of complex numbers and the familiar properties of the addition of real numbers The properties in Theorem 24 are collectively called the properties of a commutative group Another word for commutative is abelian So we say that C is a commutative group ak an abelian group under the operation of addition Note that can be regarded as a function from C gtlt C a C Such a function is often called a binary operation 22 Multiplication of Complex Numbers The de nition of multiplication for two complex numbers is at rst glance somewhat less straightforward than that of addition However it naturally follows De nition 25 Given two complex numbers phyl x2y2 E C we de ne their complep product to be 1791 2792 12 all2961112 291

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