Before Download

Question 1

j^2 = ?

Answer 1

j^2 = -1

Question 2

form of complex numbers (and which is real and imaginary component)

Answer 2

x + yj
where x & y are real numbers
x is the real component
y is the imaginary component

Question 3

when is a complex number real, when is it purely imaginary

Answer 3

when y = 0 the number is real.
when x = 0 the number is purely imaginary

Question 4

what does the conjugate of z look like (z bar)

Answer 4

_
z = x - yj

Question 5

1/z = 1/(x+yj) = ?

Answer 5

1/(x+yj) = (x-yj)/(x^2 + y^2)
good shorthand so you don’t have to do the math every time

Question 6

z(bar)(bar) = ?

Answer 6

= z

Question 7

when does z(bar) = z

Answer 7

z(bar) = z if and only if z is a REAL number

Question 8

when does z(bar) = -z

Answer 8

z(bar) = -z if and only if z is purely imaginary

Question 9

(z+w)(bar) = ?
(bar is over whole thing)

Answer 9

(z+w)(bar) = z(bar) + w(bar)

Question 10

(zw)(bar) = ?

Answer 10

(zw)(bar) = z(bar) * w(bar)

Question 11

(z^k)(bar) = ?

Answer 11

(z^k)(bar) = z(bar)^k
for k is a REAL number, k>= 0, (k =/= 0 if z=0)

Question 12

z + z(bar) = ?

Answer 12

z + z(bar) = 2x = 2Re(z)

Re(z) = real component of z

Question 13

z - z(bar) = ?

Answer 13

z - z(bar) = 2yj = 2Im(z)j

Im(z) = imaginary component of z

Question 14

z * z(bar) = ?

Answer 14

z * z(bar) = x^2 + y^2

Question 15

|z| = ?

Answer 15

|z| = sqrt(x^2 + y^2)

its just like pythagorean theorem

Question 16

|z| = 0 if and only if (?)

Answer 16

|z| = 0 if and only if z = 0

Question 17

|z(bar)| = ?

Answer 17

|z(bar)| = |z|

Question 18

z * z(bar) = ?

Answer 18

z * z(bar) = |z|^2

Question 19

|zw| = ?

Answer 19

|z| * |w|

Question 20

What is the Triangle Inequality?

Answer 20

|z + w| <= |z| + |w|

Question 21

z = x+ yj in polar form

Answer 21

z = r(cosA + jsinA)
where r = |z| = sqrt(x^2 + y^2) & theta is the angle between z & y=0

Question 22

the value of the angle in polar form is called what

Answer 22

its called an Argument of z

Question 23

how do you multiply complex numbers in polar form
(works in reverse for division)

Answer 23

you multiply the moduli(|z|, or r) and add the arguments(theta)

if z = r(cosA + jsinA)
and w = u(cosB = jsinB)
z*w = ru(cos(B+A) + jsin(B+A))

Question 24

z^n = ?

Answer 24

z^n = r^n(cos(nA) + jsin(nA))
if n is an integer

or you can just do the foil in regular form and you get
z^2 = x^2 - y^2 + 2xyj

Question 25

how do you find complex nth routes?

Answer 25

Let z = r(cosA + jsinA) be nonzero, and let n be a positive integer. Than the n distinct nth roots of z are given by

w = r^(1/n)(cos((A+2piK)/n) + jsin ((A+2piK)/n))
for k = 0, 1,…, n-1.

Question 26

what is z = r(cosA +jsinA) in exponential form

Answer 26

r*e^(j * A)

Question 27

what is Eulars identity

Answer 27

e^(j * pi) - 1 = 0

Question 28

what is the equation for a complex polynomial

Answer 28

a(n)/a(2) is not multiplication, its the number for a

z is a complex number
a(n) is a constant value

p(z) = a(n)z^n + a(n-1)z^(n-1) . . . + a(1)*z + a

Question 29

what does the addition of complex polynomials look like

Answer 29

p uses A
q uses B
p(z) + q(z) = (a(n) + b(n))z^n + …… (a(1) + b(1))z +(a(0) + b(0))

the number next to a and b represent their number, not multiplication

Question 30

if some complex number w is a rout of a real polynomial, what is another route

Answer 30

_
w , w(bar), the conjugate of w. (all mean same thing)

Question 31

what is the norm of a real vector x = [x(1), x(2), . . . x(n)] in a cartesian plane with n dimensions

Answer 31

||x|| = sqrt(x(1)^2 + x(2)^2 . . . + x(n)^2)

Question 32

what would the linear combination of the vector

[3, -5] look like as a linear combination of
e(1) = [1, 0] & e(2) = [0, 1]

Answer 32

[3, -2] = 3e(1) - 5e(2)

Question 33

what does x have to equal for ||x|| >= 0 to be true

Answer 33

x = 0 (zero vector)

Question 34

||c*x|| = ? c is a constant x is a vector

Answer 34

|c| * ||x||

Question 35

when are two vectors orthogonal

Answer 35

if x * y = 0

Question 36

the norm of a complex vector z (||z||) is equal to what

Answer 36

sqrt(z1 * z1(bar) + z2 * z2(bar)…… + zn * zn(bar))

n is equal to the degree of the vector

Question 37

what is the standard inner product for < z , w>

Answer 37

z1(bar)w1 + . . . + zn (bar)wn (bar is on first term)

Question 38

dot product of the complex number z & w

Answer 38

z * w = z1w1 + . . . + znwn
&
z * w = ||z|| * ||w|| * cos(theta)

Question 39

< z, z > >= 0 if ?

Answer 39

z = 0

Question 40

||z||^2 = ?

Answer 40

< z , z >

Question 41

< z, w >(bar) = ?

Answer 41

< w, z >(note positions are swapped)

Question 42

< v + w, z >
all 3 are complex vectors

Answer 42

< v, z > + < w, z > (z expands in)

Question 43

a < z, w > = ?

a(bar) < z, w > = ?

a is a constant

Answer 43

a < z, w > = < z, aw >

a(bar) = < az, w >

Question 44

what is the cauchy-schwarz inequality

Answer 44

|< z*w >| <= ||z|| * ||w|| (dot product)

Question 45

cross product of real vectors a & b

a x b

Answer 45

a x b = [ a2b3 - b2a3,
a3b1 - b3a1,
a1b2 - b1a2]

Question 46

where is the cross product specific to

Answer 46

cross product is specific to the real number plane in 3 dimensions

Question 47

for a x b (cross product), what is the direction of the resultant vector relative to a & b

Answer 47

the resultant vector is orthogonal to both a & b

Question 48

what is a x 0(vector) equal to

Answer 48

a x 0(vector) = 0(vector)

Question 49

a x a (cross product) = ?

Answer 49

a x a = 0(vector)

Question 50

-( a x b) = ? (cross product)

Answer 50

= b x a (positions swapped)

Question 51

||a x b|| = ? cross product

Answer 51

= ||a|| * ||b|| * sin(theta)

Dot product uses cosine

theta is the angle between the 2 vectors

Question 52

From the dot product how do you know if the vectors form an accute or obtuse angle or are orthogonal

Answer 52

=0: orthogonal
>0: acute (Positive)
<0: obtuse (negative)