Chapter 1

Question 1

|z| = √(zz)

Answer 1

False. |z| = √(zzbar)

Question 2

z1 and z2 are positive multiples of each other iff

Answer 2

z1~z2 is real and positive

Question 3

z = 1 +3i, w = 1 - 3i then z and w are orthogonal

Answer 3

False. z ~w is not purely imaginary

Question 4

2+i and 2-i are collinear

Answer 4

True. z ~w is real and positive

Question 5

1 + i and 1 - i are orthogonal

Answer 5

True. z ~w is purely imaginary

Question 6

dim C as a vector space over R

Answer 6

2

Question 7

dim C as a vector space over C is

Answer 7

1

Question 8

dim C^2 as a vector space over C is

Answer 8

2

Question 9

dim C^2 as a vector space over R is

Answer 9

4

Question 10

z = 1 + 2i, w = 2+i. inner product over C is

Answer 10

3i

Question 11

z = 1 + 2i, w = 2+i. inner product over R^2 is

Answer 11

4

Question 12

in the R^2 view of C, conjugation is equivalent to application of the linear map with matrix ((-1 0), (0, -1))

Answer 12

False. matrix is ((1 0), (0 -1))

Question 13

How are _{C} and _{R^2} related?

Answer 13

_{R^2} = Re( _{C} )

Question 14

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?

Answer 14

0. Since a 1-D vector space in C is just a line and hence all points in the vector space C are collinear.

Question 15

If z1 z2~ = Re(z1z2~) then z1 and z2 are orthogonal.

Answer 15

False

Question 16

If z1z2~ = Im(z1z2~) then z1 and z2 have the same direction.

Answer 16

False

Question 17

If z1z2~ = Re(z1z2~) > 0, then z1 and z2 have the same direction.

Answer 17

True

Question 18

|z^2| = |z|^2

Answer 18

True

Question 19

z1 = (√3/2 + i 1/2). Then z1^2 = ?

Answer 19

1/2 + i√3/2

Question 20

z1 = 2(√3/2 + i 1/2). Then z1^2 = ?

Answer 20

4(1/2 + i√3/2)

Question 21

z~ and 1/z point in the same direction

Answer 21

True

Question 22

z~ and z^2 point in the same direction. Prove or give counter example

Answer 22

False

Question 23

z~ and -z point in the same direction

Answer 23

False

Question 24

De Moivre’s Formula

Answer 24

z^n = r^n (cos n\theta + i sin n\theta)

Question 25

cos(A+B) =

Answer 25

cosAcosB - sinAsinB

Question 26

sin(A+B) =

Answer 26

sinAcosB + cosAsinB

Question 27

z^1/n =

Answer 27

r^1/n [ cos (ϴ + 2kπ)/n + i sin (ϴ + 2kπ)/n]