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1
Q

j^2 = ?

A

j^2 = -1

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2
Q

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

A

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

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3
Q

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

A

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

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4
Q

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

A

_
z = x - yj

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5
Q

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

A

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

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6
Q

z(bar)(bar) = ?

A

= z

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7
Q

when does z(bar) = z

A

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

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8
Q

when does z(bar) = -z

A

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

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9
Q

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

A

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

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10
Q

(zw)(bar) = ?

A

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

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11
Q

(z^k)(bar) = ?

A

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

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12
Q

z + z(bar) = ?

A

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

Re(z) = real component of z

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13
Q

z - z(bar) = ?

A

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

Im(z) = imaginary component of z

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14
Q

z * z(bar) = ?

A

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

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15
Q

|z| = ?

A

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

its just like pythagorean theorem

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16
Q

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

A

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

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17
Q

|z(bar)| = ?

A

|z(bar)| = |z|

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18
Q

z * z(bar) = ?

A

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

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19
Q

|zw| = ?

A

|z| * |w|

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20
Q

What is the Triangle Inequality?

A

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

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21
Q

z = x+ yj in polar form

A

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

22
Q

the value of the angle in polar form is called what

A

its called an Argument of z

23
Q

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

A

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))

24
Q

z^n = ?

A

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

25
how do you find complex nth routes?
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.
26
what is z = r(cosA +jsinA) in exponential form
r*e^(j * A)
27
what is Eulars identity
e^(j * pi) - 1 = 0
28
what is the equation for a complex polynomial
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
29
what does the addition of complex polynomials look like
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
30
if some complex number w is a rout of a real polynomial, what is another route
_ w , w(bar), the conjugate of w. (all mean same thing)
31
what is the norm of a real vector x = [x(1), x(2), . . . x(n)] in a cartesian plane with n dimensions
||x|| = sqrt(x(1)^2 + x(2)^2 . . . + x(n)^2)
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]
[3, -2] = 3e(1) - 5e(2)
33
what does x have to equal for ||x|| >= 0 to be true
x = 0 (zero vector)
34
||c*x|| = ? c is a constant x is a vector
|c| * ||x||
35
when are two vectors orthogonal
if x * y = 0
36
the norm of a complex vector z (||z||) is equal to what
sqrt(z1 * z1(bar) + z2 * z2(bar)…… + zn * zn(bar)) n is equal to the degree of the vector
37
what is the standard inner product for < z , w>
z1(bar)*w1 + . . . + zn (bar)*wn (bar is on first term)
38
dot product of the complex number z & w
z * w = z1*w1 + . . . + zn*wn & z * w = ||z|| * ||w|| * cos(theta)
39
< z, z > >= 0 if ?
z = 0
40
||z||^2 = ?
< z , z >
41
< z, w >(bar) = ?
< w, z >(note positions are swapped)
42
< v + w, z > all 3 are complex vectors
< v, z > + < w, z > (z expands in)
43
a < z, w > = ? a(bar) < z, w > = ? a is a constant
a < z, w > = < z, aw > a(bar) = < az, w >
44
what is the cauchy-schwarz inequality
|< z*w >| <= ||z|| * ||w|| (dot product)
45
cross product of real vectors a & b a x b
a x b = [ a2*b3 - b2*a3, a3*b1 - b3*a1, a1*b2 - b1*a2]
46
where is the cross product specific to
cross product is specific to the real number plane in 3 dimensions
47
for a x b (cross product), what is the direction of the resultant vector relative to a & b
the resultant vector is orthogonal to both a & b
48
what is a x 0(vector) equal to
a x 0(vector) = 0(vector)
49
a x a (cross product) = ?
a x a = 0(vector)
50
-( a x b) = ? (cross product)
= b x a (positions swapped)
51
||a x b|| = ? cross product
= ||a|| * ||b|| * sin(theta) Dot product uses cosine theta is the angle between the 2 vectors
52
From the dot product how do you know if the vectors form an accute or obtuse angle or are orthogonal
=0: orthogonal >0: acute (Positive) <0: obtuse (negative)