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Unformatted text preview: K“ x / A Set 1: Basics I Wu“ 1 10 f : 3 y : ,4 “5‘2 2 Basic Operations; Kyle A. Gallivan 11 '2 3” Department of Mathematics 3" "L ' Z “l 23" I 6 33/ 2 ‘12 ,_ 50 -104 6 Florida State University Linear Combination: Foundations of Computational Math 1 32 ‘all 2010 237 T M 2 wfi \ j \ / 1 4 WEE K fl r “tars“; Vectors and Matrices Scalarx‘ and their operations are assumed to be from 1 0 u the field of real numbers (R) 61 = 0 82 = 1 n the field of complex numbers (C) 0 0 « complex number: a : l} + 17 where 1' here is used to represent the root of 71 (occasionally we will use j for this but it will be made clear when 0 1 [MS is done) 83 : 0 e = l - ti and e, arc the real and imaginary parts of a respectively 1 1 ~ com lcx can ugaic d :13 ~ 1 P J c ’7 — the absolute value ()le denoted la] is «air : l/fi? + 77 / W /_ W Sealars, Vectors and M trices' Definition 1.1. An m X 1:, matrix of scalars from R or C is a two—dimensionally V‘“ _‘~V. ,,' cc ‘ , ,V v at 7 a \LLlUI’ Is all one dimensionally ordered list of 71 real scalars ordered arrangcmcnt of mu sealers ~ addition of vectors is componentwisc scalar addition _ l , (1H 012 Ozn — scalar vcc tor product multiplies each component of the vector With the scalar Q31 0'22 , , (12,, _ A r o C” ,, a vector is an one«dirrtcnsionally ordered list of 7: complex scalars « addition of vectnrx‘ is componentwisc complex scalar addition “'1“ a“? ’ ' ' “m" A scalar vector )ruduct multi lies each com lex com mm of the vector . ‘ , _ E p p p0 The set of m x n maU'lCCS With scalar elements from R xx denoted Rm” With the complex scalar The set of m x n, matrices with scalar elements from C is denoted C'“ "" X J K / ’; atrix Operations' Matrix scaling A, B 6 Km“ and "y E R: B ;: 7A :: 47 has elements Li] : Wag Matrix addition :1, B\ C E Rmx": C :2: A ~ B : B ~+r A has elements 71'] : Bi] + 0,] operations Matrix Vector Product ' Definition 1.2. If X A:(a1 a2 a")ERmn and the vector J‘ E R” {t 62 .l‘ : {n then Ar, : aiEi + 11252 + > > , + anEn This is the collection of vectors Rm" and the associated scalar field and / [Km—ix Operations I lfA € 52’” “‘2. B 6; R'wxwfihcn C E 52""“3 is Scalar definintion: AB has elements 7;, : 2:1Qik/3k] Matrix-vector definition: C Ab’ ,_, t; : Ab. 22:! ,,,,, 72.3 where ct : Ce;. 1), 2 Be, Outer product tictinition: ’2 v.21 C : .413 z )T ”,5? where at : Aer, (if _ eTB inner product detinmtion: k C' 2 AB hats elements “1:; :. [13-h] where b.- r Be.. (2,7- : ETA Matrix Operamgl - the matrix product is not commutative o the matrix product is associative - the matrix product is distributive, iier, A(B + C) : AB + AC 0 All scalars and vectors can be replaced with submatrices of appropriate dimension to yield block forms of the matrix product ghj Matrix Operations Definition 1.3. The transpose of A E Rm”, denoted AT, and the hemiitian transpose of A E C” x" , denoted AH . are the n X m matrices an 021 amt out Get dmt . CH2 H22 "' 017112 (312 522 ~' Elm) 7 H A : V . . A 2 an. (12,, w an”, 6m. 62,. ~ , (“rm K Vector Space I Definition 14. Given scalars .75. a set of vectors V. a vector addition operation I : y +» z for 1:, y. z E, V. and a scalarvvector product operation y : or for I. y E V and a 43 IF, we have a vector space ifthe followmg properties hold: I, + y z y -4— 1' (ll tr+y§+z : r»i‘~(_y+z) (2') x t- O“ r ,r (3) r+ (—1,): : (lv (4) (at/3)! = MM) (5) (a +2 ,5)! = oz: +51: (6) ()(I + y) : (u- + (13/ (7') 1,1 1 r (8} / lar and Vector 0 O,‘ ~ art»(---l)a prom 3 M r (71% WW8 : (O+l}o+(‘l)a scular0+1:l : (0a + la) + («1)a propfi : 0n + (1n + (71)a) prop? : 011 + (a + (71)a) propS : ()0, + (0) prop4 : 0a propfi lfl fl Examples I o P" ~ the set of polynomials of degree less than or equal to n — isomorphic to CHI — elements can be written as a linear combination of 71. + l monomials therefore finite dimensional space u 77x 7 the set of polynomials of any degree — any element can be written as a time sum of monomials - infinite dimensional since it is not the same finite sum size for all vectors . maria] : {f ; [0,5] is R, ff flownm < 00} — infinite dimensional ~ need concept of convergence to discuss infinite linear combination that {CPFCSCHCS CflCll V‘CCKDI \ J 14 \ o The algebraic structure of a vector spaces considers: e Suha’paecs « Linear Transformations - Bases - Linear independence 0 The algebraic structure of the vector spaces R" and C" is common to all linite dimensional vector spaces We will use R" in most of our discussions but the results can be adapted to C“ and all other such vector spaces, - By definition a vector space V is closed under linear combinations, but an arbitrary subset of the space is not necessarily closod‘ eg.’ a finite set or the / set of vectors with nonncgative elements. 15 r \ Definition 1.5. A subset 5 g R" is a subspace if it is closed under linear combination, Lenny/1,12. , , Matt. 6 Sthcn for any scalars om :1‘_,,,k airi+a212+~~~+otrt~ ES and in fact the subspace is itself a vector space (and hence all of our results apply within 8). Definition 1.6. Let S L: R" be a subset (finite or infinite). The set of all linear combinations of vectors in S is called the span ofS and is a subspace. Example 1.]. R" : spanieheg, _ V V , en) k _/ l6 Matrices and Transformations Definition 1.7. Given A E C"""",consider b : AI for all 1‘ E C". a The span of the columns of A is a subspace of C'" called the range MA and is denoted R(A). - Since Am: + fly) : 41,417 + flAy. A defines a linear function FM) : C” a RM) C; C“ c Any linear function F : C“ —v C" has a unique A defining it. l7 Definition 1.8. The set of vectors .771. V . . , n are linearly independent if om + - + mu : 0 M a, : 0 for L r l. , , , Ax if this does not hold then the vectors are linearly dependent. Note that: o A set of vectors being linearly dependent implies one of the vectors can be written as a linear combination of the others e Any set that contains the 0 vector is linearly dependent, x J [8 l l f ’ l y z 1 1 0 are linearly independent in El 1 l 3 J' i 1 y : 1 z : 5 1 0 1 are linearly depc ndcnt l9 / ”fl Bases Definition 1.9. A set of vectors 11,172, , . V ,n 6 S Q R" is a basis for the subspace S if - .it, , 3:1, _ , . in are linearly independent, - £[Ja71(171,112., Wm.) : .5 Note that: o A subspace has many buses but every basis contains k vectors and the unique integer k is the dimension of the subspace (k : dzm(5))_ o l: : (ltmiS) is the number of degrees of freedom in 5, ie., 5 is essentially R“ embedded in R'V . Any collection of vectors in S with k + 1 or more vectors is linearly \ ‘ ____/ dependent l Matrix Implications I o Linear independent columns ofA 6 me" a VI 75 0, Ar, ;4 0 0 Linear dependent columns ofA E Cm“ H 33* ¢ 0 3 AI, : O 0 MIA) 2 {I E C’flxlr 2 0} is a subspace called the null space of AV (Also called the kernel denoted ker(A),l o # of independent columns immersion of 7104) : column rank of A o {i of independent rows Tdnnension ofR(A) : row rank of A - lib _ A): F; Rl/l] and mnk(A) : n then the linear function defined by A is «meteoric and onto RM) and z is unique, \_____~_____/ 21 / \ Analytic Properties I ln addition to the algebraic properties discussed so far we can also define analytic propenies of veetor spaces and the associated linear transformations, - size . distance o angle These are analyzed via: . norms o inner products \ J Size and Distance Definition 1.10. A vector norm, HIM . is a function C” -r, R that satisfies 0 HI” 2 O and a: : 0 ~'-> “I“ : 0 (definiteness) . limit : lolllrll <homogeneny> ~ llr + in 2 Hill + llyll (mangle inequality) We can also deduce HI 7 yll 2 WIN , llylll mples Vector Norms I Let I (—1 C" With elements ef’r : {.3 [Norm Efivalence I Theorem 1.1. Let M1) and 1/(r) be vector norms then there exist L’OIZSILIIILY, Le, independent aft, 0 > 0 and T > 0 such that 0/41,“) g 21(2) 3 7311(1) 25 Norm Equivalence I In other words‘ for analytical purposes, all norms are equivalent. Convergence in one vector norm implies convergence in any other. Note that a and T may be dependent on 71. “I'll: S HIHI S filll‘lh [Woo S HIH2 S fillrlloo Hxlloo S MINI E "rllIlloo 26 L Matrix Norms | Definition 1.1L A matrix norm on me" denoted llAll maps €mx" ,-., R and satisfies ' EH 0 EMA“ :: fiat - 5m + Bil w: HBli Examples of matrix norms I Let A 6 CM” with elements LII/rifle, : 041'. Mix 1211a; Zilrianl : £13; HA6; n y _ H HAllw lg?" Eprlaul e 12123,, He. HAM: ' “gag! MAIN: ”Ally ' “gala-:1 “Allin ll! Alil HAHF STABLHUUV : \/ ELM/18:“? Examples of matrix norms o The Frobenius norm HAN]: is essentially the vector the matrix as if it was :1 element of CW”, 2 norm applied to 0 IA 2 : trace AHA; where the trace is the sum of the dia onal l g elements. a While all matrix norms are equivalent for analytical purposes‘ they differ considerably in their ease of computation, Matrix 2 Norm I 0 Definition given requires optimization “All; r: max “Arm llxllzzl . HAM can be related to eigenvalues and singular val also “infinite" computations can be used for approximation [MR2 S x/HAHtllAlioc k ties but these are I Bounds can be derived in terms of “Alix and EMHN. ire“ equivalence ___./ ent Matrix Norms Definition 1.12. The matrix norms H , Hm“ v “[1,” , EL, are consistent if llABllu S llAllullBlh whenever the product exists: Lemma 1.1. The matrix 11-horm defines afamt'ly ofconsistent matrix norms. Specificallij xi E C'"X",B 6 CM” and: E C“ llABllp S llAllpllBlln llA'Illp S llAllpllIllp 31 Induced Matrix Norms Definition 1.13. The matrix norm H , H is subordinate to vector norms H ‘ Ho, and il > lira if llrh'lla S llAllllIllts and the matrix norm therefore bounds the expansion/contraction of the linear transtormution defined by A, Definition 1.14. Given vector norms H » H0 and H A ”3 me induced matrix norm ll ' ”an ‘5 HAHN : Ilirliigl llAIllB \_________________/ 32 /—__~—“—\ Induced Matrix Norms ' Theorem 13. Given a Vector norm fl ~ llo on C" or R" the induced matrix norm for an n, x n, mam} Elm (subardinalc) tlollB‘iln (.mbnwllipiicazive) x] / Convergence I Both vector sequences and matrix sequences can therefore be said to converge to limit vectors and limit matrices by considering convergence in IR, Definition 1.15 For the vector sequence {It} and the matrix sequence {At} 11m xk : ;r H lim flrk 11E! : O k‘wxy kam Componentwise convergence for both follows. Angles in n-dimensional Spaces I Definition 1.16. An inner product (or scalar product) on a vector space V is a map < 1,: >2 V x V ., F where the field F is either R orC mat satisfies lv <acc+i3ziy >: a <I,y > +I3< z.y>,WifiII,y‘Z E Vand at} E F.(lineu1-ity) 2. < guy >: < 11,1 > (hermitian) 3‘ < 1.1 >2 0 and < 1,: >: 0 H x : (J (definiteness) 35 FInEéFWEEY - < 11y >2: IHy is an inner product for C” u < .13. y >2 rTy is an inner product for R" 0 There are other inner products for C” and R". . “I“ ;: v’< 1.1 > isanorm, Lemma 1.4. For I. y E C" o iJiflyé i glrllyljyéjq wnh l; + i : 1 (Hoelderinequality) c gxflyl f jrligllyfiz (Cauchy~Sc/zwarz inequality} - My; < ‘11, ally/Hm Angles can be defined by making the Cauchy-Schwinn inequality an equality, Definition 1.17. Let a: and y be two nonzero vectoxs in C" then the cosine ofthe angle between the one~dimensional Spaces defined by the vectors, 0 g (9 g 7r/2, is defined Wit/l = 6059HIHnHz/H2 \ / 37 Generalization from R2 f—fi Consider 3:. y E R2 positive quadrant. fly : cosélllwllllyll iT'g : C059 .i; z((;osl)1xsin61)andfli[l:l 1] : (cos 62,si1102) and Hg)“ :1 where 61 and 62 are angles from (1, 0) ng :: (:05 ()1 (10562 + sin 61 sin 02 : cos(0] ~ (92) : 0050 p_____________J 3B *-_\ Orthogonality Definition 1.18. The vectors .r and y am said to be orthogonal if their inner product is 0, it: , <: r. y >: rHy .: 0. This generalizes the Pythagorean Theorem to multidimensional and complex vectors: 1, , I 2 g , I! , , :zr + W: ,, (I + y) (I + til :2: J‘HI + yHy + 2Relzuy) : VEHI+yuy A Iii”: + lit/Hi Theorem 15. Let V be a vector space over IR (similar statmenlx can be made for C) with and inner product < 1:, y >. Ifthe norm is defined byllril : lr< LBthen we have ' lll‘ + UH2 :1 ”I“? + [lyllz + 2 < Ivy > ° 111 , HIV : 111112 + Hyll2 , 2 < Ly > (L‘osinelavv) - Hr + M? + HI 7 W2 : MIN? + llyllzl (parallelogram law} 0 i1]; + yHQ 711$ v sz : 4 < Ly > (polarization identity) \___________/ [Polarization and Parallelograms' Amazingly, the reverse is also true, Theorem 1.6. Let V be a normed vector space over [R (similar xtutments can be made for (C). If the norm satixfies the parallelogram law then the polarization identity defines an inner productfor V, That is ill + 9H? + [11‘ , yll) : 9(ll3‘ll2 +Hyl12) ll < m >: inn + y“: 7 HI - W} U [136+ W = ll'IH2 + “all2 + 2 < Ivy > and HI , y“? : NIH2 + Hyll2 — 2 < ray > 41 ...
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