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1 SECTION 8.4 COMPLEX VECTOR SPACES AND INNER PRODUCTS COMPLEX VECTOR SPACES AND INNER PRODUCTS All the vector spaces we have studed thus far n the text are real vector spaces snce the scalars are real numbers. A complex vector space s one n whch the scalars are complex numbers. Thus, f v 1, v 2,..., v m are vectors n a complex vector space, then a lnear combnaton s of the form where the scalars c are complex numbers. The complex verson of R n 1, c 2,..., c s the complex vector space m consstng of ordered ntuples of complex numbers. Thus, a vector n has the form c 1 v 1 c 2 v 2 It s also convenent to represent vectors n v a 1 b 1 a 2 b 2. a n b n. c m v m v a 1 b 1, 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 1 Vector Operatons n Let v 1 2, 3 and u 2, 4 be vectors n the complex vector space C 2, and determne the followng vectors. (a) v u (b) 2 v (c) 3v 5 u Soluton (a) In column matrx form, the sum v u s 1 2 v u (b) Snce and 2 3 7, we have 2 v 2 1 2, 3 5, 7. (c) 3v 5 u 3 1 2, 3 5 2, 4 (c) 3 6, , 2 4 (c) 12, 11
2 456 CHAPTER 8 COMPLEX VECTOR SPACES Many of the propertes of R n are shared by. For nstance, the scalar multplcatve dentty s the scalar 1 and the addtve dentty n s,,,...,. The standard bass for s smply e 1 1,,,..., e 2, 1,,...,.. e n,,,..., 1 R n whch s the standard bass for. Snce 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 we demonstrate n Example 2. EXAMPLE 2 Verfyng a Bass Show that v 1 v 2 v 3 S,,,,,,,, s a bass for C 3. Soluton Snce C 3 has a dmenson of 3, the set v 1, v 2, v 3 wll be a bass f t s lnearly ndependent. To check for lnear ndependence, we set a lnear combnaton of the vectors n S equal to as follows. c 1,, c 2, c 2,,, c 3,, Ths mples that { { { c 1 c 2 c 2 c 3. c 1 v 1 c 2 v 2 c 3 v 3,, c 1 c 2, c 2, c 3,, Therefore, c 1 c 2 c 3, and we conclude that v 1, 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 457 Soluton By wrtng v c 1 v 1 c 2 v 2 c 3 v 3 c 1 c 2, c 2, c 3 2,, 2 we obtan c 1 c 2 2 c 2 c 3 2 whch mples that c 2 1 and c and Therefore, c 3 2 v 1 2 v 1 v 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 where and are realvalued functons of a real varable. The set of complex numbers form the scalars for S and vector addton s defned by 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, we let c a b be a complex number. Then s n S. f x) f 1 x f 2 x f 1 f 2 f x g x f 1 x f 2 x g 1 (x g 2 x f 1 x g 1 x f 2 x g 2 x. cf x a b f 1 x f 2 x af 1 x bf 2 x bf 1 x af 2 x
4 458 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 1 v 1 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 Soluton Determne the Eucldean nner product of the vectors u 2,, 4 5 and v 1, 2,. u v u 1 v 1 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 and let k be a complex number. Then the followng propertes are true. 1. u v v u 2. u v w u w v w 3. ku v k u v 4. u kv k u v 5. u u 6. u u f and only f u. Proof We prove the frst property and leave the proofs of the remanng propertes to you. Let u u 1, u 2,..., u n and v v 1, v 2,..., v n. Then v u v 1 u 1 v 2 u 2... v n u n v 1 u 1 v 2 u 2... v n u n v 1 u 1 v 2 u 2... v n u n
5 SECTION 8.4 COMPLEX VECTOR SPACES AND INNER PRODUCTS 459 u 1 v 1 u 2 v 2 u v. u n v n We 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 1 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 1 2 u 2 2 u n d u, v u 1 v 1 2 u 2 v 2 2 u n v n EXAMPLE 6 Fndng the Eucldean Norm and Dstance n Determne the norms of the vectors u 2,, 4 5 and and fnd the dstance between u and v. v 1, 2, Soluton The norms of u and v are gven as follows. u u 1 2 u 2 2 u v v 1 2 v 2 2 v The dstance between u and v s gven by d u, v u v 1, 2,
6 46 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, we 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. 1. 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 1, u 2 v v 1, v 2. Show that the functon defned by u, v u 1 v 1 2u 2 v 2 s a complex nner product. Soluton We verfy the four propertes of a complex nner product as follows. 1. v, u v 1 u 1 2v 2 u 2 u 1 v 1 2u 2 v 2 u, v 2. u v, w u 1 v 1 w 1 2 u 2 v 2 w 2 u 1 w 1 2u 2 w 2 v 1 w 1 2v 2 w 2 u, w v, w 3. ku, v ku 1 v 1 2 ku 2 v 2 k u 1 v 1 2u 2 v 2 k u, v 4. u, u u 1 u 1 2u 2 u 2 u u 2 2 Moreover, u, u f and only f u 1 u 2. Snce all propertes hold, u, v s a complex nner product.
7 SECTION 8.4 EXERCISES 461 SECTION 8.4 EXERCISES In Exercses 1 8, perform the ndcated operaton usng u, 3, v 2, 3, and w 4, u 2. 4w w 4. v 3w 5. u 2 v v 2 2 w 7. u v 2w 8. 2v 3 w u In Exercses 9 12, determne whether S s a bass for. 9. S 1,,, 1 1. S 1,,, S,,,,,,,, S 1,, 1, 2,, 1, 1, 1, 1 In Exercses 13 16, express v as a lnear combnaton of the followng bass vectors. (a),,,,,,,, (b) 1,,, 1, 1,,,, v 1, 2, 14. v 1, 1, v, 2, v,, In Exercses 17 24, determne the Eucldean norm of v. 17. v, 18. v 1, 19. v 3 6, 2 2. v 2 3, v 1, 2, 22. v,, In Exercses 25 3, determne the Eucldean dstance between u and v v 1 2,, 3, 1 v 2, 1, 2, 4 u 1,, v, u 2, 4,, v 2, 4, u, 2, 3, v, 1, u 2, 2,, v,, u 1,, v, 1 u 1, 2, 1, 2, v, 2,, 2 In Exercses 31 34, determne whether the set of vectors s lnearly ndependent or lnearly dependent ,,, , 1, 1,,, 1, 2, 1, 33. 1,, 1,,,,,, , 1,, 1,,,, 1, 1 In Exercses 35 38, determne whether the gven functon s a complex nner product, where u u 1, u 2 and v v 1, v u, v 4u 1 v 1 6u 2 v u, v u 1 v 1 u 2 v Let v 1,, and v 2,,. If v 3 z 1, z 2, z 3 and the set {v 1, v 2, v 3 } s not a bass for C 3, what does ths mply about z 1, z 2, and z 3? 4. Let v 1,, and v 2 1,, 1. Determne a vector v 3 such that v s a bass for C 3 1, v 2, v 3. In Exercses 41 45, prove the gven property where u, v, and w are vectors n and k s a complex number. 41. u v w u w v w u kv k u v u u f and only f u. 46. 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 u, v u 1 u 2 v 2 u, v u 1 v 1 2 u 2 v 2 T 1, 2, 1, T, 1, T, ) 2, 1, T,, A 1 A A 1 A, v 1 1, v , v 1, w 1 1 2, v 5 1, w 1 1 1, w 2 2, w 2 3 ku) u v k u v u 53. Fnd the kernel of the lnear transformaton gven n Exercse 49.
8 462 CHAPTER 8 COMPLEX VECTOR SPACES 54. Fnd the kernel of the lnear transformaton gven n Exercse 5. In Exercses 55 and 56, fnd the mage of v, for the ndcated composton, where and are gven by the followng matrces. A 1 T 1 and T 2 A T 2 T T 1 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. 58. Determne whch of the followng sets are subspaces of the vector space of complexvalued functons (see Example 4). (a) The set of all functons f satsfyng f. (b) The set of all functons f satsfyng f 1. (c) The set of all functons f satsfyng f f. 8.5 UNITARY AND HERMITIAN MATRICES Problems nvolvng dagonalzaton of complex matrces, and the assocated egenvalue problems, requre the concept of untary and Hermtan matrces. These matrces roughly correspond to orthogonal and symmetrc real matrces. In order to defne untary and Hermtan matrces, we frst ntroduce the concept of the conjugate transpose of a complex matrx. Defnton of the Conjugate Transpose of a Complex Matrx The conjugate transpose of a complex matrx A, denoted by A*, s gven by A* A T where the entres of A are the complex conjugates of the correspondng entres of A. Note that f A s a matrx wth real entres, then A* A T. To fnd the conjugate transpose of a matrx, we frst calculate the complex conjugate of each entry and then take the transpose of the matrx, as shown n the followng example. EXAMPLE 1 Fndng the Conjugate Transpose of a Complex Matrx Determne A* for the matrx A
8.5 UNITARY AND HERMITIAN MATRICES. The conjugate transpose of a complex matrix A, denoted by A*, is given by
6 CHAPTER 8 COMPLEX VECTOR SPACES 5. Fnd the kernel of the lnear transformaton gven n Exercse 5. In Exercses 55 and 56, fnd the mage of v, for the ndcated composton, where and are gven by the followng
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