math 102 final review part 8!!!!!

Polynomials of a Complex Variable

100

What does the Fundamental Theorem of Algebra state?

Every polynomial of degree n has exactly n roots in C (counting multiplicities)

200

If p(z) is a polynomial with real coefficients and p(w) = 0 for some complex number w, what must be true?

The complex conjugate of w must also be a root

300

What are roots of unity?

A complex number that results in 1 when it is raised to a positive integer power

400

How do you solve for zⁿ = 1?

Compute w_★ = e^i(2pi/n) where the n distinct solutions of zⁿ = 1 are w_★, w_★², ..., w_★ⁿ

500

How do you solve zⁿ = w?

1) Find the polar form of w = |w|e^iΦ 
2) Write the particular solution z₀ = ⁿ√|w| e^iΦ/n
3) Compute w_★ = e^i(2pi/n)
4) The n distinct solutions of zⁿ = w are z₀w_★, z₀w_★², ..., z₀w_★ⁿ

Linear Algebra over Complex Numbers I

100

Given z = [z₁, z₂, ..., z_n] and w = [w₁, w₂, ..., w_n] ∈ ℂⁿ, what do the following properties equal to?

1) z + w
2) wz for any w ∈ ℂ
3) Re(z)
4) Im(z)
5) z̄

1) [z₁ + w₁, z₂ + w₂, ..., z_n + w_n]
2) [wz₁, wz₂, ..., wz_n]
3) [Re(z₁), Re(z₂), ..., Re(z_n)]
4) [Im(z₁), Im(z₂), ..., Im(z_n)]
5) [z̄₁, z̄₂, ..., z̄_n]

200

What is 1/i equal to?

-i

300

If λ is a complex eigenvalue of a real n x n matrix, then what must be true?

Its complex conjugate must also be an eigenvalue