Lecture11

# Lecture11 - 64 Lecture 11 Complex Vector Spaces 11.1...

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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.

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Lecture11 - 64 Lecture 11 Complex Vector Spaces 11.1...

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