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9 - CHAPTER 9 Complex Vector Spaces Vector Spaces over C So...

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CHAPTER 9 Complex Vector Spaces Section 9.1 Vector Spaces over C So far, we have been working entirely with vector spaces over R . However, one can define a vector space where the scalars are taken from any one system of numbers such that addition, subtraction, multiplication and division are defined for any pair of numbers (excluding di- vision by 0), and satisfy the usual commutative, associative, and distributive rules. We now look at what happens when we pick our scalars to be complex numbers. In such cases, we say that we have a vector space over C or a complex vector space. Let C 2 = { ( z 1 , z 2 ) | z 1 , z 2 C } with addition defined by EXERCISE 1 ( z 1 , z 2 ) + ( w 1 , w 2 ) = ( z 1 + w 1 , z 2 + w 2 ) , and scalar multiplication by α C defined by α ( z 1 , z 2 ) = ( αz 1 , αz 2 ) . Show that C 2 is a vector space. It is instructive to look carefully at the ideas of basis and dimension for complex vector spaces. We begin by considering the set of complex numbers C itself as a vector space. As a vector space over the complex numbers, C has a basis consisting of a single element { 1 } . That is, every complex number can be written in the form α 1 where α is a complex number. Thus, with respect to this basis, the coordinate of the complex number z is z itself. Alternatively, we could choose to use the basis { i } . Then the coordinate of z would be - iz since z = ( - iz ) i. In either case, we see that C has a basis consisting of one element, so C is a one-dimensional complex vector space. However, we could also view C as a vector space over R . Addition of complex numbers is defined as usual, and multiplication of z = x + iy by a real scalar k gives kz = kx + kyi. Observe that if we use real scalars, then the elements 1 and i in C are linearly independent. Hence, viewed as a vector space over R , the set of complex numbers is two-dimensional with ”standard” basis { 1 , i } .
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2 Chapter 9 Complex Vector Spaces We sometimes write complex numbers in a way that exhibits the property that C is a two- dimensional real vector space: we write a complex number z in the form z = x + iy = ( x, y ) = x (1 , 0) + y (0 , 1) . Note that (1 , 0) denotes the complex number 1 and that (0 , 1) denotes the complex number i . With this notation, we see that the set of complex numbers is isomorphic to R 2 , and we speak of the ”complex plane”. However, notice that this representation of the complex numbers as a real vector space does not include multiplication by complex scalars. By similar arguments to those above, we see that C 2 is a two dimensional complex vector space, but may be viewed as a real vector space of dimension four. The vector space C n is defined to be the set C n = { ( z 1 , z 2 , . . . , z n ) | z k C , k = 1 , 2 , . . . , n } , with addition of vectors and scalar multiplication defined as above. These vector spaces play an important role in much of modern mathematics.
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