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Chapter 10: Complex vector spaces Section 10.1: Complex numbers Definition: A complex number is an ordered pair of real numbers, denoted (a, b) or a + bi, where i2 = −1. If z = a + bi then the real part of z is a (Re(z) = a) and the imaginary part of z is b (Im(z) = b). Note: We can think of complex numbers geometrically as a point or vector in R2 where the real part is the x coordinate, and the imaginary part is the y coordinate. Definition: Two complex numbers a + bi, c + di are equal, written a + bi = c + di if a = c and b = d. Arithmetic with complex numbers: Addition: (a + bi) + (c + di) = (a + c) + (b + d)i Subtraction: (a + bi) − (c + di) = (a − c) + (b − d)i Scalar multiplication: k(a + bi) = (ka) + (kb)i Multiplication: (a + bi)(c + di) = ac + bdi2 + adi + bci = (ac − bd) + (ad + bc)i Example: 1 Section 10.2: Division of complex numbers Definition: The complex conjugate of a complex number z = a + bi, denoted z̄, is defined as z̄ = a − bi Geometrically, this is the reflection of z across the real axis. Definition: The modulus, or absolute value, of a complex number z = a + bi, denoted |z|, is defined as √ |z| = a2 + b2 Theorem 10.2.1: For any complex number z, zz̄ = |z|2 Theorem 10.2.2: (Revised) If z2 6= 0, then z̄2 1 = z2−1 = z2 |z2 |2 and z1 z1 z̄2 z1 z̄2 = = z2 |z2 |2 z2 z̄2 Theorem 10.2.3: Properties of the conjugate For any complex numbers z1 , z2 , 1. z1 + z2 = z̄1 + z̄2 2. z1 − z2 = z̄1 − z̄2 3. z1 z2 = z̄1 z̄2 2 4. z1 /z2 = z̄1 /z̄2 5. z̄ = z Example: 3 Section 10.3: Polar form of a complex number Definition: If z = a + bi is a complex number, then the polar form of z is written as z = r cos θ + r sin θi = r(cos θ + i sin θ) where r = |z| and θ, called the argument of z, is the angle form the positive real axis to the vector z. If −π < θ ≤ π then θ is called the principal argument of z. Note: The polar form is also called the CIS form Arithmetic using polar forms If z1 = r1 (cos θ1 + i sin θ1 ), z2 = r2 (cos θ2 + i sin θ2 ), then z1 z2 = r1 r2 (cos(θ1 + θ2 ) + i sin(θ1 + θ2 )) z1 r1 = (cos(θ1 − θ2 ) + i sin(θ1 − θ2 )) z2 r2 Example: 4 Powers of complex numbers using polar forms If z = r(cos θ + i sin θ), then z n = r n (cos(nθ) + i sin(nθ)) If r = 1 then (cos θ + i sin θ)n = cos(nθ) + i sin(nθ), DeMoivre’s formula and z 1/n = √ n θ 2kπ + r cos n n Example: 5 + i sin θ 2kπ + n n Notation: We can also write complex numbers as exponentials: z = r(cos θ + i sin θ) = reiθ Then z̄ = re−iθ and eiθ = e−iθ Example: 6 Section 10.4: Complex vector spaces Definition: A complex vector space is a vector space whose scalars can be complex numbers. In a complex vector space, a vector w ~ is a linear combination of the vectors ~v1 , . . . , ~vr if it can be written as w ~ = k1~v1 + k2~v2 + · · · + kr~vr where k1 , k2 , . . . , kr are complex numbers. The definitions of linear independence, spanning, basis, dimension and subspace are unchanged in complex vector spaces. Definition: The space of n-tuples of complex numbers, denoted Cn , has vectors of the form ~u = (u1 , u2, . . . , un ) where u1 = a1 + b1 i, u2 = a2 + b2 i, . . .. Note: The standard basis for Cn is the same as the standard basis for Rn . Example: 7 Definition: If ~u = (u1 , u2, . . . , un ), ~v = (v1 , v2 , . . . , vn ) are vectors in Cn , then their complex Euclidean inner product is defined as ~u · ~v = u1 v̄1 + u2 v̄2 + · · · + un v̄n where v̄j is the complex conjugate of vj . Theorem 10.4.1: Properties of the complex inner product If ~u, ~v , w ~ are vectors in Cn , k is any complex number, then 1. ~u · ~v = ~v · ~u 2. (~u + ~v ) · w ~ = ~u · w ~ + ~v · w ~ 3. (k~u) · ~v = k(~u · ~v) 4. ~v · ~v ≥ 0, ~v · ~v = 0 if and only if ~v~0 Definition: Euclidean norm of the vector ~u = (u1 , u2 , . . . , un ) in Cn is defined as k~uk = (~u · ~u)1/2 p |u1 |2 + |u2 |2 + · · · + |un |2 = Definition: Euclidean distance between two vectors ~u, ~v in Cn is defined as d(~u, ~v) = k~u − ~v k p |u1 − v1 |2 + |u2 − v2 |2 + · · · + |un − vn |2 = Example: 8