8.4 COMPLEX VECTOR SPACES AND INNER PRODUCTS


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1 SECTION 8.4 COMPLEX VECTOR SPACES AND INNER PRODUCTS COMPLEX VECTOR SPACES AND INNER PRODUCTS All the vector spaces you have studed thus far n the text are real vector spaces because the scalars are real numbers. A complex vector space s one n whch the scalars are complex numbers. So, f v, v 2,..., v m are vectors n a complex vector space, then a lnear combnaton s of the form c v c 2 v 2 where the scalars c are complex numbers. The complex verson of R n, c 2,..., c s the complex vector space m consstng of ordered ntuples of complex numbers. So, a vector n has the form It s also convenent to represent vectors n v a b a 2 b 2. a n b n. c m v m v a b, a 2 b 2,..., a n b n. by column matrces of the form As wth R n, the operatons of addton and scalar multplcaton n are performed component by component. EXAMPLE Vector Operatons n Let v 2, 3 and u 2, 4 be vectors n the complex vector space C 2. Determne each vector. (a) v u (b) 2 v (c) 3v 5 u Soluton (a) In column matrx form, the sum v u s v u (b) Because and 2 3 7, you have 2 v 2 2, 3 5, 7. (c) 3v 5 u 3 2, 3 5 2, 4 (c) 3 6, , 2 4 (c) 2,
2 494 CHAPTER 8 COMPLEX VECTOR SPACES Many of the propertes of R n are shared by. For nstance, the scalar multplcatve dentty s the scalar and the addtve dentty n s,,,...,. The standard bass for s smply whch s the standard bass for R n. Because ths bass contans n vectors, t follows that the dmenson of s n. Other bases exst; n fact, any lnearly ndependent set of n vectors n can be used, as demonstrated n Example 2. e,,,..., e 2,,,...,.. e n,,,..., EXAMPLE 2 Verfyng a Bass Show that S,,,,,,,, s a bass for C 3. Soluton Because C 3 has a dmenson of 3, the set v, v 2, v 3 wll be a bass f t s lnearly ndependent. To check for lnear ndependence, set a lnear combnaton of the vectors n S equal to as follows. c,, c 2, c 2,,, c 3,, Ths mples that c c 2 v v 2 v 3 c 2 c 3. c v c 2 v 2 c 3 v 3,, c c 2, c 2, c 3,, So, c c 2 c 3, and you can conclude that v, v 2, v 3 s lnearly ndependent. EXAMPLE 3 Representng a Vector n by a Bass Use the bass S n Example 2 to represent the vector v 2,, 2.
3 SECTION 8.4 COMPLEX VECTOR SPACES AND INNER PRODUCTS 495 Soluton By wrtng you can obtan whch mples that c 2 and So, v c v c 2 v 2 c 3 v 3 c c 2, c 2, c 3 2,, 2, c c 2 2 c 2 c 2 c and c 3 2 v 2 v v 2 2 v Try verfyng that ths lnear combnaton yelds 2,, 2. Other than, there are several addtonal examples of complex vector spaces. For nstance, the set of m n complex matrces wth matrx addton and scalar multplcaton forms a complex vector space. Example 4 descrbes a complex vector space n whch the vectors are functons. EXAMPLE 4 The Space of ComplexValued Functons Consder the set S of complexvalued functons of the form f x) f x f 2 x where f and f 2 are realvalued functons of a real varable. The set of complex numbers form the scalars for S and vector addton s defned by f x g x f x f 2 x g (x g 2 x f x g x f 2 x g 2 x. It can be shown that S, scalar multplcaton, and vector addton form a complex vector space. For nstance, to show that S s closed under scalar multplcaton, let c a b be a complex number. Then cf x a b f x f 2 x af x bf 2 x bf x af 2 x s n S.
4 496 CHAPTER 8 COMPLEX VECTOR SPACES The defnton of the Eucldean nner product n s smlar to that of the standard dot product n R n, except that here the second factor n each term s a complex conjugate. Defnton of Eucldean Inner Product n Let u and v be vectors n. The Eucldean nner product of u and v s gven by u v u v u 2 v 2 u n v n. REMARK: Note that f u and v happen to be real, then ths defnton agrees wth the standard nner (or dot) product n. R n EXAMPLE 5 Fndng the Eucldean Inner Product n C 3 Determne the Eucldean nner product of the vectors u 2,, 4 5 and v, 2,. Soluton u v u v u 2 v 2 u 3 v Several propertes of the Eucldean nner product are stated n the followng theorem. Theorem 8.7 Propertes of the Eucldean Inner Product Let u, v, and w be vectors n propertes are true u v v u u v w u w v w ku v k u v u kv k u v u u 6. u u f and only f u. and let k be a complex number. Then the followng Proof The proof of the frst property s gven, and the proofs of the remanng propertes have been left to you. Let u u, u 2,..., u n and v v, v 2,..., v n. Then v u v u v 2 u 2... v n u n v u v 2 u 2... v n u n v u v 2 u 2... v n u n
5 SECTION 8.4 COMPLEX VECTOR SPACES AND INNER PRODUCTS 497 u v u 2 v 2 u v. u n v n You wll now use the Eucldean nner product n to defne the Eucldean norm (or length) of a vector n and the Eucldean dstance between two vectors n. Defnton of Norm The Eucldean norm (or length) of u n s denoted by u and s and Dstance n u u u 2. The Eucldean dstance between u and v s d u, v u v. The Eucldean norm and dstance may be expressed n terms of components as u u 2 u u n 2 2 d u, v u v 2 u 2 v u n v n 2 2. EXAMPLE 6 Fndng the Eucldean Norm and Dstance n Determne the norms of the vectors u 2,, 4 5 and v, 2, and fnd the dstance between u and v. Soluton The norms of u and v are gven as follows. u u 2 u 2 2 u v v 2 v 2 2 v The dstance between u and v s gven by d u, v u v, 2,
6 498 CHAPTER 8 COMPLEX VECTOR SPACES Complex Inner Product Spaces The Eucldean nner product s the most commonly used nner product n. However, on occason t s useful to consder other nner products. To generalze the noton of an nner product, use the propertes lsted n Theorem 8.7. Defnton of a Complex Inner Product Let u and v be vectors n a complex vector space. A functon that assocates wth u and v the complex number u, v s called a complex nner product f t satsfes the followng propertes.. u, v v, u 2. u v, w u, w v, w 3. ku, v k u, v 4. u, u and u, u f and only f u. A complex vector space wth a complex nner product s called a complex nner product space or untary space. EXAMPLE 7 A Complex Inner Product Space Let u u and be vectors n the complex space C 2, u 2 v v, v 2. Show that the functon defned by u, v u v 2u 2 v 2 s a complex nner product. Soluton Verfy the four propertes of a complex nner product as follows.. v, u v u 2v 2 u 2 u v 2u 2 v 2 u, v 2. u v, w u v w 2 u 2 v 2 w 2 u w 2u 2 w 2 v w 2v 2 w 2 u, w v, w 3. ku, v ku v 2 ku 2 v 2 k u v 2u 2 v 2 k u, v 4. u, u u u 2u 2 u 2 u 2 2 u 2 2 Moreover, u, u f and only f u u 2. Because all propertes hold, u, v s a complex nner product.
7 SECTION 8.4 EXERCISES 499 SECTION 8.4 EXERCISES In Exercses 8, perform the ndcated operaton usng u, 3, v 2, 3, and w 4, 6.. 3u 2. 4w 3. 2 w 4. v 3w 5. u 2 v v 2 2 w 7. u v 2w 8. 2v 3 w u In Exercses 9 2, determne whether S s a bass for. 9. S,,,. S,,,. S,,,,,,,, 2. S,,, 2,,,,, In Exercses 3 6, express v as a lnear combnaton of each of the bass vectors. (a),,,,,,,, (b),,,,,,,, 3. v, 2, 4. v,, 3 5. v, 2, 6. v,, In Exercses 7 24, determne the Eucldean norm of v. 7. v, 8. v, 9. v 3 6, 2 2. v 2 3, v, 2, 22. v,, 23. v 2,, 3, 24. v 2,, 2, 4 In Exercses 25 3, determne the Eucldean dstance between u and v u,, v, u 2, 4,, v 2, 4, u, 2, 3, v,, u 2, 2,, v,, u,, v, u, 2,, 2, v, 2,, 2 In Exercses 3 34, determne whether the set of vectors s lnearly ndependent or lnearly dependent. 3.,,, 32.,,,,,, 2,, 33.,,,,,,,, 34.,,,,,,,, In Exercses 35 38, determne whether the functon s a complex nner product, where u u, u 2 and v v, v u, v 4u v 6u 2 v u, v u v u 2 v Let v,, and v 2,,. If v 3 z, z 2, z 3 and the set v s not a bass for C 3, v 2, v 3, what does ths mply about z, z 2, and z 3? 4. Let v,, and v 2,,. Determne a vector v 3 such that v s a bass for C 3, v 2, v 3. In Exercses 4 45, prove the gven property where u, v, and w are vectors n and k s a complex number. 4. u v w u w v w u kv k u v 44. u u 45. u u f and only f u. 46. Wrtng Let u, v be a complex nner product and k a complex number. How are u, v and u, kv related? In Exercses 47 and 48, determne the lnear transformaton T : C m that has the gven characterstcs In Exercses 49 52, the lnear transformaton T : C m s gven by T v Av. Fnd the mage of v and the premage of w. 49. A 5. A u, v u u 2 v 2 u, v u v 2 u 2 v 2 T, 2,, T,, T, ) 2,, T,, A A,, v,, v 2 3 2,, v, v 2 5, w w w ku v k u v w 53. Fnd the kernel of the lnear transformaton gven n Exercse Fnd the kernel of the lnear transformaton gven n Exercse 5.
8 5 CHAPTER 8 COMPLEX VECTOR SPACES In Exercses 55 and 56, fnd the mage of v, for the ndcated 58. Determne whch of the followng sets are subspaces of the composton, where T and T 2 are gven by the followng matrces. vector space of complexvalued functons (see Example 4). and A 2 A (a) The set of all functons f satsfyng f. (b) The set of all functons f satsfyng f. 55. T 2 T (c) The set of all functons f satsfyng f f. 56. T T Determne whch of the followng sets are subspaces of the vector space of 2 2 complex matrces. (a) The set of 2 2 symmetrc matrces. (b) The set of 2 2 matrces A satsfyng A T A. (c) The set of 2 2 matrces n whch all entres are real. (d) The set of 2 2 dagonal matrces.
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