VGLA flashcards term 2
properties of summing complex numbers
- closed
- associative
- identity (o+oi)
- inverse (-a-bi)
- commutative
properties of multiplying complex numbers
- closed
- associative
- identity
- most have an inverse (0 does not have an inverse)
- commutative
multiplying a complex number by its complex conjugate gives
a positive real number
argand diagram axes
y axis imaginary
x axis real
solving an expression to find the complex number z
- replace z with a+bi
2. rearrange and equate real and imaginary parts to find a and b
arg(z) =
arctan(y/x) + 2kπ
principle argument based on quadrants
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
argument of a real number a
0
argument of an imaginary number bi
π/2
how is eulers formula derived
taylor expansion of cosθ + isinθ and comparison to taylor expansion of eˣ
cos(θ) =
1/2(eᶦⁿθ + e⁻ᶦⁿθ)
sin(θ) =
1/2(eᶦⁿθ - e⁻ᶦⁿθ)
how to expand powers of cos or sin
- write trig in complex numbers form
- apply demoivres (or just apply straight off)
- split up into individual cosnθ’s and sinnθ’s
how to represent cos(nθ)/sin(nθ) as powers of cos or sin
- write full cos(nθ) + isin(nθ)
- expand using bionomial expansion
- equate real and imaginary parts depending of which trig function you had originally (real if cos, imaginary is sin)
finding the nth roots of a complex number
- get into eulers form
- ω = ρeᶦϕ = z¹/ⁿ => ωⁿ = z
- equate real and imaginary parts
eⁿ = |z|
nϕ = θ + k2π - sub in k=1,2,3…n to find all the principle arguments
- sub back in to find the complex number roots
how many nth roots of a complex number z are there
n
where do all the nth roots of a complex number lie
on the unit circle
what shape do the nth roots make on the argand diagram
a regular n sided polygon (equal angles)
two distinct real roots =>
b²-4ac>0
two repeated real roots =>
b²-4ac=0
a complex conjugate pair of roots =>
b²-4ac<0
dotted line
not included in loci
solid line
included in loci
determinant of a 2x2 matrix
ad-bc
determinant in summation for
sum to n from k =1 aᵢₖCᵢₖ(A)
where i is the row you are expanding along
Cᵢₖ(A) =
(-1)ᶦ⁺ᵏMᵢₖ(A)
determinant of an upper triangular/lower triangular/ diagonal matrix
the product of the diagonal
determinant of the unit matrix
1
determinant of a matrix with a row or column of zeros
0
determinant of a matrix where a row of column is duplicated
0
proving that swapping rows gives the negative determinant
- expand along row i in A
- expand along row j in B
- 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
- 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)
swapping rows when finding the determinant
det(A) = - det(B)
multiplying a row by a constant when finding the determinant
det(A) = λdet(B)
proving that multiplying a row by a constant gives that multiple of the determinant
- expand along row i in A
- expand along row i in B
- 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
- The determinants are the same except B is multiplied by lambda by the definition of a determinant
adding a multiple of another row to a row when finding the determinant
- expand along row i in A
- expand along row i in B
- 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
- hence det(A) = det(B)
For elementary matrices |Eᵢⱼ|
= -1
For elementary matrices |Eᵢλ|
= λ
For elementary matrices |Eᵢⱼλ|
= 1
determinant product theorem
det(A.B) = det(A).det(B)
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)
transposing a matrix
reflect it along leading diagonal
properties of the transpose function
- (A.B)ᵀ = Aᵀ .Bᵀ
2. |Aᵀ| = |A|
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
tan(π/3)
√3
tan(π/4)
1
tan(π/6)
√3/3
