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Unformatted text preview: 64 Lecture 11 Complex Vector Spaces 11.1 Introduction Up to now we have only considered vector spaces over the field of real num bers. For the Fourier transform we will need vector spaces where we can multiply vectors by complex numbers. If x ∈ R , then we know that x 2 ≥ and x 2 = 0 only if x = 0. Therefore there is no real number such that x 2 = 1, or: There is no real solution to the equation x 2 + 1 = 0 . More generally let us look at the equation x 2 + bx + c = 0 By completing the square we get x = b 2 ± 1 2 √ b 2 4 c There are now 3 possibilities: 1. b 2 4 c > 0. Then the quadratic equation above gives two solutions x = b 2 + 1 2 √ b 2 4 c and x = b 2 1 2 √ b 2 4 c 65 66 LECTURE 11. COMPLEX VECTOR SPACES 2. b 2 4 c = 0. Then we have one solution x = b 2 3. b 2 4 c < 0. Then there is no real solution to quadratic equation x 2 + bx + c = 0. 11.2 Complex numbers We now introduce a new number i = √ 1 such that i 2 = 1 The complex numbers are all expressions of the form z = x + iy , x,y ∈ R . The set of complex numbers is denoted by C . We say that x = < z is the real part of z and y = = z is the imaginary part of z . Recall that the set of real numbers can be thought of as a line, the real line . To picture the set of complex numbers we use the plane. A vector ~v = ( x,y ) corresponds to the complex number z = x + iy . 11.2.1 Addition and multiplication on C The addition of two complex numbers z = x + iy and w = s + it then corresponds to the addition of the corresponding vectors. Thus ( x + iy ) + ( s + it ) = ( x + s ) + i ( y + t ) . To find out what the product of z and w is, we use the familiar rules along with i 2 = 1. Thus ( x + iy ) · ( s + it ) = xs + xit + iys + iyit = xs + i 2 yt + i ( xt + ys ) = ( xs yt ) + i ( xt + ys ) . 11.2. COMPLEX NUMBERS 67 11.2.2 Conjugate and absolute value of z Before we find the inverse or reciprocal of z = x + iy , we need to introduce the complex conjugate: x + iy = x iy Thus, complex conjugate corresponds to a reflection around the...
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This note was uploaded on 11/23/2011 for the course MATH 2025 taught by Professor Staff during the Spring '08 term at LSU.
 Spring '08
 Staff
 Real Numbers, Vectors, Vector Space, Complex Numbers

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