man 2007/2/16 page 234 CHAPTER 4 Vector Spaces To crtcze mathematcs for ts abstracton s to mss the pont entrel. Abstracton s what makes mathematcs work. Ian Stewart The man am of ths tet s to stud lnear mathematcs. In Chapter 2 we studed sstems of lnear equatons, and the theor underlng the soluton of a sstem of lnear equatons can be consdered as a specal case of a general mathematcal framework for lnear problems. To llustrate ths framework, we dscuss an eample. Consder the homogeneous lnear sstem A = 0, where 1 1 2 A = 2 2 4. 3 3 6 It s straghtforward to show that ths sstem has soluton set S = {(r 2s, r, s) : r, s R}. Geometrcall we can nterpret each soluton as definng the coordnates of a pont n space or, equvalentl, as the geometrc vector wth components v = (r 2s, r, s). Usng the standard operatons of vector addton and multplcaton of a vector b a real number, t follows that v can be wrtten n the form v = r(1, 1, 0) + s( 2, 0, 1). We see that ever soluton to the gven lnear problem can be epressed as a lnear combnaton of the two basc solutons (see Fgure 4.0.1): v1 = (1, 1, 0) and v2 = ( 2, 0, 1). 234
4.1 Vectors n R n 235 3 v 2 ( 2, 0, 1) v rv 1 + sv 2 2 1 v 1 (1, 1, 0) Fgure 4.0.1: Two basc solutons to A = 0 and an eample of an arbtrar soluton to the sstem. We wll observe a smlar phenomenon n Chapter 6, when we establsh that ever soluton to the homogeneous second-order lnear dfferental equaton can be wrtten n the form + a 1 + a 2 = 0 () = c 1 1 () + c 2 2 (), where 1 () and 2 () are two nonproportonal solutons to the dfferental equaton on the nterval of nterest. In each of these problems, we have a set of vectors V (n the frst problem the vectors are ordered trples of numbers, whereas n the second, the are functons that are at least twce dfferentable on an nterval I) and a lnear vector equaton. Further, n both cases, all solutons to the gven equaton can be epressed as a lnear combnaton of two partcular solutons. In the net two chapters we develop ths wa of formulatng lnear problems n terms of an abstract set of vectors, V, and a lnear vector equaton wth solutons n V. We wll fnd that man problems ft nto ths framework and that the solutons to these problems can be epressed as lnear combnatons of a certan number (not necessarl two) of basc solutons. The mportance of ths result cannot be overemphaszed. It reduces the search for all solutons to a gven problem to that of fndng a fnte number of solutons. As specfc applcatons, we wll derve the theor underlng lnear dfferental equatons and lnear sstems of dfferental equatons as specal cases of the general framework. Before proceedng further, we gve a word of encouragement to the more applcatonorented reader. It wll probabl seem at tmes that the deas we are ntroducng are rather esoterc and that the formalsm s pure mathematcal abstracton. However, n addton to ts nherent mathematcal beaut, the formalsm ncorporates deas that pervade man areas of appled mathematcs, partcularl engneerng mathematcs and mathematcal phscs, where the problems under nvestgaton are ver often lnear n nature. Indeed, the lnear algebra ntroduced n the net two chapters should be consdered an etremel mportant addton to one s mathematcal repertore, certanl on a par wth the deas of elementar calculus. 4.1 Vectors n R n In ths secton, we use some famlar deas about geometrc vectors to motvate the more general and abstract dea of a vector space, whch wll be ntroduced n the net secton. We begn b recallng that a geometrc vector can be consdered mathematcall as a drected lne segment (or arrow) that has both a magntude (length) and a drecton attached to t. In calculus courses, we defne vector addton accordng to the parallelogram law (see Fgure 4.1.1); namel, the sum of the vectors and s the dagonal of
236 CHAPTER 4 Vector Spaces the parallelogram formed b and. We denote the sum b +. It can then be shown geometrcall that for all vectors,, z, Fgure 4.1.1: Parallelogram law of vector addton. and + = + (4.1.1) + ( + z) = ( + ) + z. (4.1.2) These are the statements that the vector addton operaton s commutatve and assocatve. The zero vector, denoted 0, s defned as the vector satsfng + 0 =, (4.1.3) for all vectors. We consder the zero vector as havng zero magntude and arbtrar drecton. Geometrcall, we pcture the zero vector as correspondng to a pont n space. Let denote the vector that has the same magntude as, but the opposte drecton. Then accordng to the parallelogram law of addton, + ( ) = 0. (4.1.4) k, k 0 k, k 0 Fgure 4.1.2: Scalar multplcaton of b k. The vector s called the addtve nverse of. Propertes (4.1.1) (4.1.4) are the fundamental propertes of vector addton. The basc algebra of vectors s completed when we also defne the operaton of multplcaton of a vector b a real number. Geometrcall, f s a vector and k s a real number, then k s defned to be the vector whose magntude s k tmes the magntude of and whose drecton s the same as f k > 0, and opposte to f k<0. (See Fgure 4.1.2.) If k = 0, then k = 0. Ths scalar multplcaton operaton has several mportant propertes that we now lst. Once more, each of these can be establshed geometrcall usng onl the foregong defntons of vector addton and scalar multplcaton. For all vectors and, and all real numbers r, s and t, 1 =, (4.1.5) (st) = s(t), (4.1.6) r( + ) = r + r, (4.1.7) (s + t) = s + t. (4.1.8) It s mportant to realze that, n the foregong development, we have not defned a multplcaton of vectors. In Chapter 3 we dscussed the dea of a dot product and cross product of two vectors n space (see Equatons (3.1.4) and (3.1.5)), but for the purposes of dscussng abstract vector spaces we wll essentall gnore the dot product and cross product. We wll revst the dot product n Secton 4.11, when we develop nner product spaces. We wll see n the net secton how the concept of a vector space arses as a drect generalzaton of the deas assocated wth geometrc vectors. Before performng ths abstracton, we want to recall some further features of geometrc vectors and gve one specfc and mportant etenson. We begn b consderng vectors n the plane. Recall that R 2 denotes the set of all ordered pars of real numbers; thus, R 2 ={(, ) : R, R}. The elements of ths set are called vectors n R 2, and we use the usual vector notaton to denote these elements. Geometrcall we dentf the vector v = (, ) n R 2 wth
(0, ) v (, ) (, 0) Fgure 4.1.3: Identfng vectors n R 2 wth geometrc vectors n the plane. 4.1 Vectors n R n 237 the geometrc vector v drected from the orgn of a Cartesan coordnate sstem to the pont wth coordnates (, ). Ths dentfcaton s llustrated n Fgure 4.1.3. The numbers and are called the components of the geometrc vector v. The geometrc vector addton and scalar multplcaton operatons are consstent wth the addton and scalar multplcaton operatons defned n Chapter 2 va the correspondence wth row (or column) vectors for R 2 : If v = ( 1, 1 ) and w = ( 2, 2 ), and k s an arbtrar real number, then v + w = ( 1, 1 ) + ( 2, 2 ) = ( 1 + 2, 1 + 2 ), (4.1.9) kv = k( 1, 1 ) = (k 1,k 1 ). (4.1.10) These are the algebrac statements of the parallelogram law of vector addton and the scalar multplcaton law, respectvel. (See Fgure 4.1.4.) Usng the parallelogram law of vector addton and Equatons (4.1.9) and (4.1.10), t follows that an vector v = (, ) can be wrtten as v = + j = (1, 0) + (0, 1), where = (1, 0) and j = (0, 1) are the unt vectors pontng along the postve - and -coordnate aes, respectvel. ( 1 2, 1 2 ) ( 2, 2 ) w v w v ( 1, 1 ) kv (k 1, k 1 ) Fgure 4.1.4: Vector addton and scalar multplcaton n R 2. The propertes (4.1.1) (4.1.8) are now easl verfed for vectors n R 2. In partcular, the zero vector n R 2 s the vector 0 = (0, 0). Furthermore, Equaton (4.1.9) mples that (, ) + (, ) = (0, 0) = 0, so that the addtve nverse of the general vector v = (, ) s v = (, ). It s straghtforward to etend these deas to vectors n 3-space. We recall that R 3 ={(,,z): R, R,z R}. As llustrated n Fgure 4.1.5, each vector v = (,,z)n R 3 can be dentfed wth the geometrc vector v that jons the orgn of a Cartesan coordnate sstem to the pont wth coordnates (,,z). We call,, and z the components of v.
238 CHAPTER 4 Vector Spaces z (0, 0, z) (,, z) v (0,, 0) (, 0, 0) (,, 0) Fgure 4.1.5: Identfng vectors n R 3 wth geometrc vectors n space. Recall that f v = ( 1, 1,z 1 ), w = ( 2, 2,z 2 ), and k s an arbtrar real number, then addton and scalar multplcaton were gven n Chapter 2 b v + w = ( 1, 1,z 1 ) + ( 2, 2,z 2 ) = ( 1 + 2, 1 + 2,z 1 + z 2 ), (4.1.11) kv = k( 1, 1,z 1 ) = (k 1,k 1,kz 1 ). (4.1.12) Once more, these are, respectvel, the component forms of the laws of vector addton and scalar multplcaton for geometrc vectors. It follows that an arbtrar vector v = (,,z)can be wrtten as v = + j + zk = (1, 0, 0) + (0, 1, 0) + z(0, 0, 1), where = (1, 0, 0), j = (0, 1, 0), and k = (0, 0, 1) denote the unt vectors whch pont along the postve -, -, and z-coordnate aes, respectvel. We leave t as an eercse to check that the propertes (4.1.1) (4.1.8) are satsfed b vectors n R 3, where 0 = (0, 0, 0), and the addtve nverse of v = (,,z)s v = (,, z). We now come to our frst major abstracton. Whereas the sets R 2 and R 3 and ther assocated algebrac operatons arse naturall from our eperence wth Cartesan geometr, the motvaton behnd the algebrac operatons n R n for larger values of n does not come from geometr. Rather, we can vew the addton and scalar multplcaton operatons n R n for n>3as the natural etenson of the component forms of addton and scalar multplcaton n R 2 and R 3 n (4.1.9) (4.1.12). Therefore, n R n we have that f v = ( 1, 2,..., n ), w = ( 1, 2,..., n ), and k s an arbtrar real number, then v + w = ( 1 + 1, 2 + 2,..., n + n ), (4.1.13) kv = (k 1,k 2,...,k n ). (4.1.14) Agan, these defntons are drect generalzatons of the algebrac operatons defned n R 2 and R 3, but there s no geometrc analog when n>3. It s easl establshed that these operatons satsf propertes (4.1.1) (4.1.8), where the zero vector n R n s 0 = (0, 0,...,0), and the addtve nverse of the vector v = ( 1, 2,..., n ) s v = ( 1, 2,..., n ). The verfcaton of ths s left as an eercse.
4.1 Vectors n R n 239 Eample 4.1.1 If v = (1.2, 3.5, 2, 0) and w = (12.23, 19.65, 23.22, 9.76), then v + w = (1.2, 3.5, 2, 0) + (12.23, 19.65, 23.22, 9.76) = (13.43, 23.15, 25.22, 9.76) and 2.35v = (2.82, 8.225, 4.7, 0). Eercses for 4.1 Ke Terms Vectors n R n, Vector addton, Scalar multplcaton, Zero vector, Addtve nverse, Components of a vector. Sklls Be able to perform vector addton and scalar multplcaton for vectors n R n gven n component form. Understand the geometrc perspectve on vector addton and scalar multplcaton n the cases of R 2 and R 3. Be able to formall verf the aoms (4.1.1) (4.1.8) for vectors n R n. True-False Revew For Questons 1 12, decde f the gven statement s true or false, and gve a bref justfcaton for our answer. If true, ou can quote a relevant defnton or theorem from the tet. If false, provde an eample, llustraton, or bref eplanaton of wh the statement s false. 1. The vector (, ) n R 2 s the same as the vector (,, 0) n R 3. 2. Each vector (,,z) n R 3 has eactl one addtve nverse. 3. The soluton set to a lnear sstem of 4 equatons and 6 unknowns conssts of a collecton of vectors n R 6. 4. For ever vector ( 1, 2,..., n ) n R n, the vector ( 1) ( 1, 2,..., n ) s an addtve nverse. 5. A vector whose components are all postve s called a postve vector. 6. If s and t are scalars and and are vectors n R n, then (s + t)( + ) = s + t. 7. For ever vector n R n, the vector 0 s the zero vector of R n. 8. The parallelogram whose sdes are determned b vectors and n R 2 have dagonals determned b the vectors + and. 9. If s a vector n the frst quadrant of R 2, then an scalar multple k of s stll a vector n the frst quadrant of R 2. 10. The vector 5 6j + 2k n R 3 s the same as (5, 6, 2). 11. Three vectors,, and z n R 3 alwas determne a 3-dmensonal sold regon n R 3. 12. If and are vectors n R 2 whose components are even ntegers and k s a scalar, then + and k are also vectors n R 2 whose components are even ntegers. Problems 1. If = (3, 1), = ( 1, 2), determne the vectors v 1 = 2, v 2 = 3, v 3 = 2 + 3. Sketch the correspondng ponts n the -plane and the equvalent geometrc vectors. 2. If = ( 1, 4) and = ( 5, 1), determne the vectors v 1 = 3, v 2 = 4, v 3 = 3+( 4). Sketch the correspondng ponts n the -plane and the equvalent geometrc vectors. 3. If = (3, 1, 2, 5), = ( 1, 2, 9, 2), determne v = 5 + ( 7) and ts addtve nverse. 4. If = (1, 2, 3, 4, 5) and z = ( 1, 0, 4, 1, 2), fnd n R 5 such that 2 + ( 3) = z.
240 CHAPTER 4 Vector Spaces 5. Verf the commutatve law of addton for vectors n R 4. 6. Verf the assocatve law of addton for vectors n R 4. 7. Verf propertes (4.1.5) (4.1.8) for vectors n R 3. 8. Show wth eamples that f s a vector n the frst quadrant of R 2 (.e., both coordnates of are postve) and s a vector n the thrd quadrant of R 2 (.e., both coordnates of are negatve), then the sum + could occur n an of the four quadrants. 4.2 Defnton of a Vector Space In the prevous secton, we showed how the set R n of all ordered n-tuples of real numbers, together wth the addton and scalar multplcaton operatons defned on t, has the same algebrac propertes as the famlar algebra of geometrc vectors. We now push ths abstracton one step further and ntroduce the dea of a vector space. Such an abstracton wll enable us to develop a mathematcal framework for studng a broad class of lnear problems, such as sstems of lnear equatons, lnear dfferental equatons, and sstems of lnear dfferental equatons, whch have far-reachng applcatons n all areas of appled mathematcs, scence, and engneerng. Let V be a nonempt set. For our purposes, t s useful to call the elements of V vectors and use the usual vector notaton u, v,...,to denote these elements. For eample, f V s the set of all 2 2 matrces, then the vectors n V are 2 2 matrces, whereas f V s the set of all postve ntegers, then the vectors n V are postve ntegers. We wll be nterested onl n the case when the set V has an addton operaton and a scalar multplcaton operaton defned on ts elements n the followng senses: Vector Addton: A rule for combnng an two vectors n V. We wll use the usual + sgn to denote an addton operaton, and the result of addng the vectors u and v wll be denoted u + v. Real (or Comple) Scalar Multplcaton: A rule for combnng each vector n V wth an real (or comple) number. We wll use the usual notaton kv to denote the result of scalar multplng the vector v b the real (or comple) number k. To combne the two tpes of scalar multplcaton, we let F denote the set of scalars for whch the operaton s defned. Thus, for us, F s ether the set of all real numbers or the set of all comple numbers. For eample, f V s the set of all 2 2 matrces wth comple elements and F denotes the set of all comple numbers, then the usual operaton of matr addton s an addton operaton on V, and the usual method of multplng a matr b a scalar s a scalar multplcaton operaton on V. Notce that the result of applng ether of these operatons s alwas another vector (2 2 matr) n V. As a further eample, let V be the set of postve ntegers, and let F be the set of all real numbers. Then the usual operatons of addton and multplcaton wthn the real numbers defne addton and scalar multplcaton operatons on V. Note n ths case, however, that the scalar multplcaton operaton, n general, wll not eld another vector n V, snce when we multpl a postve nteger b a real number, the result s not, n general, a postve nteger. We are now n a poston to gve a precse defnton of a vector space.