VGLA Theorems term 2

Question 1

multiplication in mod-arg

Answer 1

given any two no-zero complex numbers z₁ z₂

i) |z₁z₂| = |z₁||z₂|
(ii) arg(z₁ z₂) = arg(z₁) + arg(z₂

Question 2

division in mod-arg

Answer 2

given any two non-zero complex numbers z₁ z₂

i) |z₁/z₂| = |z₁|/|z₂|
(ii) arg(z₁/z₂) = arg(z₁) - arg(z₂

Question 3

De moivres theorem

Answer 3

if n∈ℕ of θ is real then:

cos(θ) + isin(θ))ⁿ = cos(nθ) + isin(nθ

Question 4

Eulers formula

Answer 4

z = re^iθ

Question 5

factors of polynomials

Answer 5

The complex number α is a root of the polynomial equation p(z)=0 if and only if z-α is a factor of the polynomial p(z)

Question 6

pair of complex roots

Answer 6

Any non-real roots of the polynomial equation p(z)=0 of degree n occur in complex conjugate pairs
(polynomial has real coefficiants)

Question 7

how factors are expressed

Answer 7

A polynomial p(z) can be expressed as a product of real factors in one of the following ways

• product of linear factors only
• product of irreducible quadratic factors only
• product of atleast one linear equation and atleast one irreducible quadratic factor

Question 8

number of factors

Answer 8

A polynomial p(z) of degree n with real coefficiants has exactly n zeros, some of which may be complex conjugate pairs

Question 9

fundamental theorem of algebra

Answer 9

Any polynomial equation of degree n with n>=1 with complex coefficiants has atleast one complex root

Question 10

|z-z₀| = α

Answer 10

The set of solutions to the equation is represented on the argand diagram by the points on a circle with centre z₀ and radius α

Question 11

Re(α) / Im(α)

Answer 11

the set of solutions re[resented on the argand diagram by the points on the line where all the real parts are α (vertical line) or all the imaginary parts are α (horizontal line)

Question 12

Arg(z-z₀) = θ

Answer 12

Set of solutions are represented on the argand diagram by a half line from z₀ at the angle θ

Question 13

|z-(a+bi)|=|z-(c+di)|

Answer 13

Set of soltuions are represented on the argand diagram by the perpendicular bisector between the points a+bi and c+di

Question 14

Expansion of any row or column

Answer 14

For any matrix A=[aᵢⱼ]∈Mₙₙ.

det(A) = aᵢ₁Cᵢ₁(A) + aᵢ₂Cᵢ₂(A) + … + aᵢₙCᵢₙ(A) for any 1<i></i>

Question 15

determinant of an upper triangular matrix

Answer 15

product of the diagonal

Question 16

determinant of a lower triangular matrix

Answer 16

product of the diagonal

Question 17

determinant of a diagonal matrix

Answer 17

product of the diagonal

Question 18

determinant of the unit matrix

Answer 18

1

Question 19

determinant of a matrix where there is a row or column of 0

Answer 19

0

Question 20

determinant where there is a repeated row or column

Answer 20

0

Question 21

interchanging rows/column when finding the determinant

Answer 21

det(P) = - det(A)

Question 22

multiplying a row/column by a constant λ when finding the determinant

Answer 22

det(P) = λdet(A)

Question 23

adding a multiple of another row/column when finding the determinant

Answer 23

det(P) = det(A)

Question 24

Elementary matrices multiples of determinants

Answer 24

det(E.A) = det(E) . det(A)

Question 25

non-invertible matrices

Answer 25

If the matrix A is non-invertible then det(A)=0

Question 26

Determinant product theorem

Answer 26

Given the matrices A and B then det(AB) = det(A)det(B)

Question 27

invertible matrices

Answer 27

A matrix A is invertible if and only if det(A)!=0. When det(A)!=0 det(A⁻¹)= (det(A))⁻¹ = 1/det(A)

Question 28

no trivial solution corollary

Answer 28

A homogeneous system A.x=0 of n equations in n unknowns has a non-trivial solution if and only if det(A) = 0

Question 29

Expansion of transposition

Answer 29

Given matrices A,B Then (A.B)ᵀ = Aᵀ.Bᵀ

Question 30

Transpose of an elementary matrix

Answer 30

Given an elementary matrix E then |Eᵀ|=|E|

Question 31

determinant of transpose matrix

Answer 31

For a matrix A det(Aᵀ) = det(A)

Question 32

replacement of summed rows

Answer 32

For a matrix A=[aᵢⱼ]∈Mₙₙ and where aₚⱼ=bₚⱼ+cₚⱼ for 1<=p<=n then:
det(A) = det(B) +det(C)
where the patrix B is constructed by replacing row p in A by the values bₚⱼ and he matrix C is constructed by replacing row p in A with the values Cₚⱼ

Question 33

Inverse of a matrix

Answer 33

For a matrix A∈Mₙₙ A.adj(A) = adj(A).A = det(A).Iₙ in particular when det(A)!=0 A⁻¹ = 1/det(A) . adj(A)

Question 34

Crammers rule

Answer 34

If A∈Mₙₙ is invertible, then the unique solution of the system A.x=b of n linear equations in n unknowns 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 35

unique identities

Answer 35

if a binary operation on a set V has an identity it is unique

Question 36

zero vector uniqueness

Answer 36

The zero vector (identity) in a real vector space ℝ,+,• is unique

Question 37

inverse of addition uniqueness

Answer 37

In a real vector space, V, the inverse with respect to the addition on V of a vector v is unique

Question 38

scalar multiplication distributivity

Answer 38

Consider m vectors in a real vector space v₁, v₂, … vₘ ∈V, with m≥2 and λ∈ℝ, then
λ(v₁ + v₂ + … + vₘ) = λv₁, λv₂, … λvₘ

Question 39

scalar multiplication distributivity (inside out)

Answer 39

Consider a vector v∈V with V real vector space and m real numbers λ₁, λ₂,..,λₖ∈ℝ, with m≥2, then
v(λ₁ + λ₂ + … + λₘ) = vλ₁, vλ₂, … vλₘ

Question 40

inverses an identities of vector spaces theoerem

0= 0 vector

Answer 40

If V is a real vector space, then

1. ∀v∈V: 0v = 0
2. ∀λ∈ℝ: λ0 = 0
3. ∀λ∈ℝ, ∀v∈V: if λv = 0 then either λ=0 (scalar) or v=0
4. ∀λ∈ℝ, ∀v∈V: (-λ)v = λ(-v) = -λv

Question 41

inverse of a vector (for addition)

Answer 41

∀v∈V: (-1)v = -v

Question 42

subspace of a vector space theorem

Answer 42

A non-empty subset I of a real vector space V is a subspace of V if and only if
1. ∀u,v∈U: u+v∈U
2. ∀u∈U, ∀λ∈ℝ: λu∈U
in other words U is a subspace of V iff U is closed under addition and scalar multiplication

Question 43

subspace of spanning sets theorem

Answer 43

Suppose that U = span{u₁, u₂, … uₖ} where u₁, u₂, … uₖ∈V with V a real vector space. Then

1. u₁, u₂, … uₖ∈U
2. U is a subspace of V
3. U is he smallest subspace of V containing each of the vectors u₁, u₂, … uₖ in the sense that if Ú is another subspace of V which contains all k of those vectors, the U⊆Ú

Question 44

row and column space of a transpose matrix

Answer 44

Consider a matrix A∈Mₘₙ. Then,

i) row(A) = col(Aᵀ
(ii) col(A) = row(Aᵀ)

Question 45

linear dependency due to scalar multiples theorem

Answer 45

A set {u₁, u₂} of two non-zero vectors in a real vector space V is linearly dependent if and only if u₂ is a scalar multiple of u₁

Question 46

linear independency due to scalar multiple

Answer 46

A set {u₁, u₂} of two non-zero vectors in a real vector space V is linearly independent if and only if u₂ is a not scalar multiple of u₁

Question 47

linear combinations and linear dependency

Answer 47

Consider an ordered set of k non-zero vectors in a real vector space V, S = {u₁, u₂, … uₖ}. The set S is linearly dependent if and only if there is a vector of the set that can be expressed as a linear combination of preceding vectors

Question 48

number of linearly indepdent sets vs spanning sets theorem

Answer 48

Let S be a set of k vectors in a real vector space V which spans V. Let T be a linearly independent set of m vectors in V. Then
m≤k
(the number of vectors in a linearly independent set in V cannot exceed the number of vectors in a spanning set of V)