University:
Whitman College
Course:
MATH-240 | Linear Algebra
Academic year:

2020
Views:

196

Pages:

1
Author:

sizwakalev84

Complex Number Summary Sheet You should read the Appendix in the Linear Algebra book (and in the Calculus book) for more details. This is meant to simply summarize the results. 1. Definition: If z = a + bi, then Re(z) = a, and Im(a) = b. √ 2. Definition: C = a + bi : a, b ∈ IR, i = −1 =Space of complex numbers (it is a vector space). 3. The real numbers are a subspace of the complex numbers. (Set b = 0 in a + bi)- That is, IR ⊂ C a + bi n 4. Definition: C is the set of n-tuples of complex numbers, i.e., ∈ C2 c + di 5. Operations on C: Let z = a + bi, w = c + di. (a) z + w = a + bi + c + di = (a + c) + (b + d)i (b) cz = ca + cbi (for c ∈ IR) (c) zw = (a + bi)(c + di) = ac + adi + bci − bd = (ac − bd) + (ad + bc)i (d) z = a + bi = a − bi (Complex conjugation) p √ √ (e) |z| = z z̄ = (a + bi)(a − bi) = a2 + b2 =Modulus (or size) of z 1 z̄ a b = = + i (f) z z z̄ a2 + b2 a2 + b2 6. Definition: The argument of z is the angle θ ∈ (−π, π] so that θ is the angle the vector (a, b)T makes with the vector (1, 0) (translated to be between −π and π). Formally,  if a > 0, b ∈ IR tan−1 ab    π  if a = 0, b > 0  2 − π2 if a = 0, b < 0 arg(z) =  −1 b  tan + π if a < 0, b > 0   a  tan−1 ab − π if a < 0, b < 0 The argument is not defined for z = 0. In Matlab, the argument of a + bi is: atan2(b,a) 7. Definition: (Euler’s Formula) eiθ = cos(θ) + i sin(θ) √ 8. The polar form of a + bi is: reiθ again where r = a2 + b2 and θ = arg(z). 2 9. Multiplication in C by z = a + bi corresponds to matrix multiplication in IR by a −b b a For example, if w = c + di in C, it corresponds to the vector [c, d]T ∈ IR2 . Now, zw in C gives: (a + bi)(c + di) = (ac − bd) + (ad + bc)i, which corresponds to the vector [ac − bd, ad + bc]T . Alternatively in IR2 , a −b c ac − bd = b a d ad + bc 1

Linear Algebra Complex Number Summary Sheet

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