{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

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

This preview shows pages 1–3. Sign up to view the full content.

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 } .

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
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.
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### What students are saying

• As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

Kiran Temple University Fox School of Business ‘17, Course Hero Intern

• I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

Dana University of Pennsylvania ‘17, Course Hero Intern

• The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

Jill Tulane University ‘16, Course Hero Intern