VGLA flashcards term 2

1
Q

properties of summing complex numbers

A
  • closed
  • associative
  • identity (o+oi)
  • inverse (-a-bi)
  • commutative
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2
Q

properties of multiplying complex numbers

A
  • closed
  • associative
  • identity
  • most have an inverse (0 does not have an inverse)
  • commutative
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3
Q

multiplying a complex number by its complex conjugate gives

A

a positive real number

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4
Q

argand diagram axes

A

y axis imaginary

x axis real

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5
Q

solving an expression to find the complex number z

A
  1. replace z with a+bi

2. rearrange and equate real and imaginary parts to find a and b

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6
Q

arg(z) =

A

arctan(y/x) + 2kπ

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7
Q

principle argument based on quadrants

A

x,y positive: tanθ = y/x
x postive, y negative: tanθ = -y/x
x negative, y negative: tanθ = y/x - π
x negative, y positive: tanθ = -y/x + π

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8
Q

argument of a real number a

A

0

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9
Q

argument of an imaginary number bi

A

π/2

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10
Q

how is eulers formula derived

A

taylor expansion of cosθ + isinθ and comparison to taylor expansion of eˣ

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11
Q

cos(θ) =

A

1/2(eᶦⁿθ + e⁻ᶦⁿθ)

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12
Q

sin(θ) =

A

1/2(eᶦⁿθ - e⁻ᶦⁿθ)

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13
Q

how to expand powers of cos or sin

A
  1. write trig in complex numbers form
  2. apply demoivres (or just apply straight off)
  3. split up into individual cosnθ’s and sinnθ’s
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14
Q

how to represent cos(nθ)/sin(nθ) as powers of cos or sin

A
  1. write full cos(nθ) + isin(nθ)
  2. expand using bionomial expansion
  3. equate real and imaginary parts depending of which trig function you had originally (real if cos, imaginary is sin)
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15
Q

finding the nth roots of a complex number

A
  1. get into eulers form
  2. ω = ρeᶦϕ = z¹/ⁿ => ωⁿ = z
  3. equate real and imaginary parts
    eⁿ = |z|
    nϕ = θ + k2π
  4. sub in k=1,2,3…n to find all the principle arguments
  5. sub back in to find the complex number roots
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16
Q

how many nth roots of a complex number z are there

A

n

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17
Q

where do all the nth roots of a complex number lie

A

on the unit circle

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18
Q

what shape do the nth roots make on the argand diagram

A

a regular n sided polygon (equal angles)

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19
Q

two distinct real roots =>

20
Q

two repeated real roots =>

21
Q

a complex conjugate pair of roots =>

22
Q

dotted line

A

not included in loci

23
Q

solid line

A

included in loci

24
Q

determinant of a 2x2 matrix

25
determinant in summation for
sum to n from k =1 aᵢₖCᵢₖ(A) | where i is the row you are expanding along
26
Cᵢₖ(A) =
(-1)ᶦ⁺ᵏMᵢₖ(A)
27
determinant of an upper triangular/lower triangular/ diagonal matrix
the product of the diagonal
28
determinant of the unit matrix
1
29
determinant of a matrix with a row or column of zeros
0
30
determinant of a matrix where a row of column is duplicated
0
31
proving that swapping rows gives the negative determinant
1. expand along row i in A 2. expand along row j in B 3. write the matrix of minors for A and B (remove the jth column in A and B when doing this) you then have the minor matrix of A = minor matrix of B 4. the determinants are the same except for the change in power on the -1 hence sign switches (this works for next door rows, however an odd number of swaps are always needed)
32
swapping rows when finding the determinant
det(A) = - det(B)
33
multiplying a row by a constant when finding the determinant
det(A) = λdet(B)
34
proving that multiplying a row by a constant gives that multiple of the determinant
1. expand along row i in A 2. expand along row i in B 3. write the matrix of minors for A and B (removing the ith column in A and B when doing this) you have the minor matrix of A = minor matrix of B 4. The determinants are the same except B is multiplied by lambda by the definition of a determinant
35
adding a multiple of another row to a row when finding the determinant
1. expand along row i in A 2. expand along row i in B 3. splitting the addition row into two matrices each with one part of the addition you have that det(B) = det(A) + det(Q) where Q is a matrix where the row j is repeated so the determinant is zero 4. hence det(A) = det(B)
36
For elementary matrices |Eᵢⱼ|
= -1
37
For elementary matrices |Eᵢλ|
= λ
38
For elementary matrices |Eᵢⱼλ|
= 1
39
determinant product theorem
det(A.B) = det(A).det(B)
40
invertible matrices theorem
``` A matrix A∈Mₙₙ is invertible if and only if det(A)!=0 when det(A⁻¹)=(det(A))⁻¹ = 1/det(A) ```
41
transposing a matrix
reflect it along leading diagonal
42
properties of the transpose function
1. (A.B)ᵀ = Aᵀ .Bᵀ | 2. |Aᵀ| = |A|
43
crammers rule
If A∈Mₙₙ is invertible, the unique of the system A.x=b of n linear equations in unknowns (where x and b are vectors) is given by x₁=det(A₁)/det(A), x₂=det(A₂)/det(A), xₙ=det(Aₙ)/det(A). For each k=1,2,...n the matrix Aₖ is obtained by replacing the entries in column k of A by the entries in the column vector b
44
tan(π/3)
√3
45
tan(π/4)
1
46
tan(π/6)
√3/3