Chapter 1 Flashcards
|z| = √(zz)
False. |z| = √(zzbar)
z1 and z2 are positive multiples of each other iff
z1~z2 is real and positive
z = 1 +3i, w = 1 - 3i then z and w are orthogonal
False. z ~w is not purely imaginary
2+i and 2-i are collinear
True. z ~w is real and positive
1 + i and 1 - i are orthogonal
True. z ~w is purely imaginary
dim C as a vector space over R
2
dim C as a vector space over C is
1
dim C^2 as a vector space over C is
2
dim C^2 as a vector space over R is
4
z = 1 + 2i, w = 2+i. inner product over C is
3i
z = 1 + 2i, w = 2+i. inner product over R^2 is
4
in the R^2 view of C, conjugation is equivalent to application of the linear map with matrix ((-1 0), (0, -1))
False. matrix is ((1 0), (0 -1))
How are _{C} and _{R^2} related?
_{R^2} = Re( _{C} )
suppose z1, z2, z3 \in C. Consider C as a 1-dimensional vector space over C. Then the distance between z3 and the line determined by z2 and z3 is?
- Since a 1-D vector space in C is just a line and hence all points in the vector space C are collinear.
If z1 z2~ = Re(z1z2~) then z1 and z2 are orthogonal.
False
If z1z2~ = Im(z1z2~) then z1 and z2 have the same direction.
False
If z1z2~ = Re(z1z2~) > 0, then z1 and z2 have the same direction.
True
|z^2| = |z|^2
True
z1 = (√3/2 + i 1/2). Then z1^2 = ?
1/2 + i√3/2
z1 = 2(√3/2 + i 1/2). Then z1^2 = ?
4(1/2 + i√3/2)
z~ and 1/z point in the same direction
True
z~ and z^2 point in the same direction. Prove or give counter example
False
z~ and -z point in the same direction
False
De Moivre’s Formula
z^n = r^n (cos n\theta + i sin n\theta)
cos(A+B) =
cosAcosB - sinAsinB
sin(A+B) =
sinAcosB + cosAsinB
z^1/n =
r^1/n [ cos (ϴ + 2kπ)/n + i sin (ϴ + 2kπ)/n]
