VGLA flashcards term 2

Question 1

properties of summing complex numbers

Answer 1

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

Question 2

properties of multiplying complex numbers

Answer 2

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

Question 3

multiplying a complex number by its complex conjugate gives

Answer 3

a positive real number

Question 4

argand diagram axes

Answer 4

y axis imaginary

x axis real

Question 5

solving an expression to find the complex number z

Answer 5

1. replace z with a+bi

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

Question 6

arg(z) =

Answer 6

arctan(y/x) + 2kπ

Question 7

principle argument based on quadrants

Answer 7

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

Question 8

argument of a real number a

Answer 8

0

Question 9

argument of an imaginary number bi

Answer 9

π/2

Question 10

how is eulers formula derived

Answer 10

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

Question 11

cos(θ) =

Answer 11

1/2(eᶦⁿθ + e⁻ᶦⁿθ)

Question 12

sin(θ) =

Answer 12

1/2(eᶦⁿθ - e⁻ᶦⁿθ)

Question 13

how to expand powers of cos or sin

Answer 13

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

Question 14

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

Answer 14

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)

Question 15

finding the nth roots of a complex number

Answer 15

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

Question 16

how many nth roots of a complex number z are there

Answer 16

n

Question 17

where do all the nth roots of a complex number lie

Answer 17

on the unit circle

Question 18

what shape do the nth roots make on the argand diagram

Answer 18

a regular n sided polygon (equal angles)

Question 19

two distinct real roots =>

Answer 19

b²-4ac>0

Question 20

two repeated real roots =>

Answer 20

b²-4ac=0

Question 21

a complex conjugate pair of roots =>

Answer 21

b²-4ac<0

Question 22

dotted line

Answer 22

not included in loci

Question 23

solid line

Answer 23

included in loci

Question 24

determinant of a 2x2 matrix

Answer 24

ad-bc

Question 25

determinant in summation for

Answer 25

sum to n from k =1 aᵢₖCᵢₖ(A)

where i is the row you are expanding along

Question 26

Cᵢₖ(A) =

Answer 26

(-1)ᶦ⁺ᵏMᵢₖ(A)

Question 27

determinant of an upper triangular/lower triangular/ diagonal matrix

Answer 27

the product of the diagonal

Question 28

determinant of the unit matrix

Answer 28

1

Question 29

determinant of a matrix with a row or column of zeros

Answer 29

0

Question 30

determinant of a matrix where a row of column is duplicated

Answer 30

0

Question 31

proving that swapping rows gives the negative determinant

Answer 31

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)

Question 32

swapping rows when finding the determinant

Answer 32

det(A) = - det(B)

Question 33

multiplying a row by a constant when finding the determinant

Answer 33

det(A) = λdet(B)

Question 34

proving that multiplying a row by a constant gives that multiple of the determinant

Answer 34

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

Question 35

adding a multiple of another row to a row when finding the determinant

Answer 35

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)

Question 36

For elementary matrices |Eᵢⱼ|

Answer 36

= -1

Question 37

For elementary matrices |Eᵢλ|

Answer 37

= λ

Question 38

For elementary matrices |Eᵢⱼλ|

Answer 38

= 1

Question 39

determinant product theorem

Answer 39

det(A.B) = det(A).det(B)

Question 40

invertible matrices theorem

Answer 40

A matrix A∈Mₙₙ is invertible if and only if det(A)!=0 
when det(A⁻¹)=(det(A))⁻¹ = 1/det(A)

Question 41

transposing a matrix

Answer 41

reflect it along leading diagonal

Question 42

properties of the transpose function

Answer 42

1. (A.B)ᵀ = Aᵀ .Bᵀ

2. |Aᵀ| = |A|

Question 43

crammers rule

Answer 43

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

Question 44

tan(π/3)

Answer 44

√3

Question 45

tan(π/4)

Answer 45

1

Question 46

tan(π/6)

Answer 46

√3/3