Elementary Linear Algebra 6e - Larson, Edwards, Falvo - Chapter 8.4

Elementary Linear Algebra 6e - Larson, Edwards, Falvo - Chapter 8.4

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8.4 COMPLEX VECTOR SPACES AND INNER PRODUCTS All the vector spaces you have studied thus far in the text are real vector spaces because the scalars are real numbers. A complex vector space is one in which the scalars are com- plex numbers. So, if are vectors in a complex vector space, then a linear combination is of the form where the scalars are complex numbers. The complex version of is the complex vector space consisting of ordered n -tuples of complex numbers. So, a vector in has the form It is also convenient to represent vectors in by column matrices of the form As with the operations of addition and scalar multiplication in are performed com- ponent by component. EXAMPLE 1 Vector Operations in C n Let and be vectors in the complex vector space Determine each vector. (a) (b) (c) Solution (a) In column matrix form, the sum is (b) Because and you have (c) (c) (c) 5 s 12 2 i , 2 11 1 i d 5 s 3 1 6 i , 9 2 3 i d 2 s 2 9 1 7 i , 20 2 4 i d 3 v 2 s 5 2 i d u 5 3 s 1 1 2 i , 3 2 i d 2 s 5 2 i ds 2 2 1 i , 4 d s 2 1 i d v 5 s 2 1 i ds 1 1 2 i , 3 2 i d 5 s 5 i , 7 1 i d . s 2 1 i ds 3 2 i d 5 7 1 i , s 2 1 i ds 1 1 2 i d 5 5 i v 1 u 5 3 1 1 2 i 3 2 i 4 1 3 2 2 1 i 4 4 5 3 2 1 1 3 i 7 2 i 4 . v 1 u 3 v 2 s 5 2 i d u s 2 1 i d v v 1 u C 2 . u 5 s 2 2 1 i , 4 d v 5 s 1 1 2 i , 3 2 i d C n R n , v 5 3 a 1 1 b 1 i a 2 1 b 2 i . . . a n 1 b n i 4 . C n v 5 s a 1 1 b 1 i , a 2 1 b 2 i , . . . , a n 1 b n i d . C n C n R n c 1 , c 2 , . . . , c m c 1 v 1 1 c 2 v 2 1 ? ? ? 1 c m v m v 1 , v 2 , . . . , v m SECTION 8.4 COMPLEX VECTOR SPACES AND INNER PRODUCTS 493
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Many of the properties of are shared by For instance, the scalar multiplicative identity is the scalar 1 and the additive identity in is The standard basis for is simply which is the standard basis for Because this basis contains n vectors, it follows that the dimension of is n. Other bases exist; in fact, any linearly independent set of n vectors in can be used, as demonstrated in Example 2. EXAMPLE 2 Verifying a Basis Show that is a basis for Solution Because has a dimension of 3, the set will be a basis if it is linearly inde- pendent. To check for linear independence, set a linear combination of the vectors in S equal to 0 as follows. This implies that So, and you can conclude that is linearly independent. EXAMPLE 3 Representing a Vector in C n by a Basis Use the basis S in Example 2 to represent the vector v 5 s 2, i , 2 2 i d . H v 1 , v 2 , v 3 J c 1 5 c 2 5 c 3 5 0, c 3 i 5 0. c 2 i 5 0 s c 1 1 c 2 d i 5 0 ss c 1 1 c 2 d i , c 2 i , c 3 i d 5 s 0, 0, 0 d s c 1 i , 0, 0 d 1 s c 2 i , c 2 i , 0 d 1 s 0, 0, c 3 i d 5 s 0, 0, 0 d c 1 v 1 1 c 2 v 2 1 c 3 v 3 5 s 0, 0, 0 d H v 1 , v 2 , v 3 J C 3 C 3 . S
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Elementary Linear Algebra 6e - Larson, Edwards, Falvo - Chapter 8.4

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