GenVecThry - 3 General Vector Spaces Much of what we did...

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Unformatted text preview: 3 General Vector Spaces Much of what we did with traditional vectors can also be done with other sets of “things”. We will develop the appropriate theory here, and extend our notation appropriately. One change we will make is that we will no longer restrict ourselves to real scalars. Hence- forth, the term “scalar” will refer to either a real number or a complex number, depending on the context. Unless otherwise indicated, the default will be that scalars are complex numbers. Since we will be using complex numbers as scalars, we will begin by briefly reviewing some of the basic ideas and notation of complex analysis. We will also briefly review the complex exponential since complex exponentials will be used in some of the examples. 3.1 Elementary Complex Analysis ∗ Basic Ideas, Notation and Terminology Recall that z is a complex number if and only if it can be written as z = x + iy where x and y are real numbers and i is a “complex constant” satisfying i 2 = − 1 . The real part of z , denoted by Re [ z ] , is the real number x , while the imaginary part of z , denoted by Im [ z ] , is the real number y . 1 If Im [ z ] = 0 (equivalently, z = Re [ z ] ), then z is said to be real. Conversely, if Re [ z ] = 0 (equivalently, z = i Im [ z ] ), then z is said to be imaginary. The complex conjugate of z = x + iy , which we will denote by z ∗ , is the complex number z ∗ = x − iy . In the future, given any statement like “the complex number z = x + iy ”, it should auto- matically be assumed (unless otherwise indicated) that x and y are real numbers. The algebra of complex numbers can be viewed as simply being the algebra of real numbers with the addition of a number i whose square is negative one. Thus, choosing some computations that will be of particular interest, zz ∗ = z ∗ z = ( x − iy )( x + iy ) = x 2 − ( iy ) 2 = x 2 + y 2 ∗ Much of this section has been stolen, slightly modified, from Principles of Fourier Analysis by Howell, with the permission of the author. It may contain a little more than we really need for now. 1 Our text uses H5228 and H5219 instead of Re and Im . 9/9/2011 General Vector Spaces Chapter & Page: 3–2 x y r θ z = x + iy Imaginary Axis Real Axis Figure 3.1: Coordinates in the complex plane for z = x + iy , where x > 0 and y > 0 . and 1 z = 1 x + iy = 1 x + iy · x − iy x − iy = x − iy x 2 + y 2 = z ∗ zz ∗ . We will often use the easily verified facts that, for any pair of complex numbers z and w , ( z + w) ∗ = z ∗ + w ∗ and ( z w) ∗ = ( z ∗ )(w ∗ ) . The set of all complex numbers is denoted by H11923 . By associating the real and imaginary parts of the complex numbers with the coordinates of a two-dimensional Cartesian system, we can identify H11923 with a plane (called, unsurprisingly, the complex plane). This is illustrated in figure 3.1. Also indicated in this figure are the corresponding polar coordinates r and θ for z = x + iy . The value r , which we will also denote by...
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GenVecThry - 3 General Vector Spaces Much of what we did...

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