VGLA definitions term 2

Question 1

Imaginary Unit

Answer 1

The Imaginary unit i is defined such that i²=-1 and (-i)²=-1

Question 2

Complex numbers

Answer 2

The set of complex numbers ℂ is defined by ℂ = {a+bi: a∈ℝ and b∈ℝ}

Question 3

equality of complex numbers

Answer 3

Two complex numbers (a+bi), (c+di) are equal if and only if

1. a=c
2. b=d

Question 4

Sum of complex numbers

Answer 4

The sum of two complex numbers a+bi and c+di is the complex number (a+bi) + (c+ di) = (a+c) + (b+d)i

Question 5

product of two complex numbers

Answer 5

The product of two complex numbers a+bi and c+di is the complex number (a+bi)(c+di) = (ac-bd)+(bc+ad)i

Question 6

real part of a complex number

Answer 6

The real part of a complex number z=a+bi is given by Re(z) =a

Question 7

Imaginary part of a complex number

Answer 7

The imaginary part of a complex number is given by Im(z)=b

Question 8

complex conjugate

Answer 8

The complex conjugate of a complex number z=a+bi is given by z*=a+bi

Question 9

modulus

Answer 9

The modulus of a complex number z=x+yi, |z| is given by |z|=√(a²+ b²)

Question 10

Argument

Answer 10

The value of θ for which x = |z|cos(θ) and y = |z|sin(θ) is an argument of the complex number z= x+yi where x!=0.

Question 11

principle argument

Answer 11

The principle value of an argument arg(z) is the value of arg(z) such that arg(z) complex numbers ℂ is defined by ℂ = {a+bi: a∈ℝ and b∈(-π,π)

Question 12

roots of complex numbers

Answer 12

a complex number ω is the nth root of a complex number z if and only if wⁿ=z

Question 13

polynomial of degree n (real coefficiants)

Answer 13

A polynomial of degree n in the indeterminate (unknown) z with real coefficiants is an expression of the form aₙzⁿ+ aₙ₋₁zⁿ⁻¹+ … + a₁z + a₀
where aᵢ∈ℝ for i=0,1,2…n and aₙ!=0

Question 14

zero

Answer 14

A complex number α is called a zero of the polynomial p(z) of degree n with real coefficiants if p(α)=0 i.e.
aₙαⁿ+ aₙ₋₁αⁿ⁻¹+ … + a₁α + a₀ = 0

Question 15

multiplicity

Answer 15

The number of times an item appears in an expression

Question 16

|z-z₀| = α

Answer 16

The loci of points on a circle with centre z₀ and radius α

Question 17

Arg(z- z₀) = θ

Answer 17

The loci of points on the half line starting at z₀ at the angle θ to the horizontal

Question 18

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

Answer 18

The loci of points on the perpendicular bisector between the points a+bi and c+di

Question 19

Submatrix

Answer 19

A submatrix of A∈Mₘₙ is obtained from A by deleting either

(i) at least one row of A
(ii) at least one column of A
(iii) at least one row and at least one column of A

Question 20

(i,j)-minor

Answer 20

The (i,j)-minor of a matrix A∈Mₙₙ written as Mᵢⱼ(A) is the determinant of the (n-1)x(n-1)-submatrix of A obtained by deleting row i and column j of A

Question 21

(i,j)-cofactor

Answer 21

The (i,j)-cofactor of a matrix A∈Mₙₙ, written as Cᵢⱼ(A), is defined by
Cᵢⱼ(A) = (-1)ᶦ⁺ʲMᵢⱼ(A)

Question 22

determinant

Answer 22

The determinant of a matrix A∈Mₙₙ is given by det(A) = a₁₁C₁₁(A) + a₁₂C₁₂(A) + … + a₁ₙC₁ₙ(A)

Question 23

Transpose Matrix

Answer 23

The transpose matrix Aᵀ = [bᵢⱼ]∈Mₘₙ of a matrix A = [aᵢⱼ]∈Mₘₙ is a matrix where the entry in position (j,i) is given by the entry in position (i,j) of the matrix A i.e.
bᵢⱼ = aᵢⱼ

Question 24

Cofactor Matrix

Answer 24

The cofactor matrix C(A) = [cᵢⱼ] of A = [aᵢⱼ]∈Mₙₙ is the matrix in Mₙₙ whos (i,j)-th entry is the cofactor of aᵢⱼ in A i.e.
cᵢⱼ = Cᵢⱼ(A)

Question 25

Adjoint Matrix

Answer 25

The adjoint matrix adj(A) of A = [aᵢⱼ]∈Mₙₙ is the transpose of the cofactor matrix of A i.e. adj(A) = (C(A))ᵀ

Question 26

binary operation

Answer 26

binary operation on a set V is a rule under which each ordered pair of two elements of V is associated with a unique element ( which may or may not belong to V)

Question 27

internal binary operation

Answer 27

An internal binary operation on a set V is a binary operation, , such that
xy ∈ V
for all x,y ∈V. The set is said to be closed under such a binary operation.

Question 28

group

Answer 28

A set V with a binary operation * is called a group under the binary operation * when the following conditions are satisfied:

1. • is an internal binary operation
2. • is associative
3. • has an identity
4. for every element in V there exists an inverse element

Question 29

Abelian group

Answer 29

A group under a binary operation * is abelian when * is commutative

Question 30

field

Answer 30

A set V with binary operations * and # is called a field V,*,# when:

1. V,* is an abelian group with identity 0
2. V{0}, # is an abelian group with identity 1
3. ∀x∈V: 0#x = x#0 = 0
4. ∀x,y,z∈V: x#(yz) = (x#y)(x#z)

Question 31

vectors space

Answer 31

A set V with a binary operation # and scalar multiplication * with elements from a field F,+,• is called a vector space over F when the following properties are satisfied:
1. V,# is an abelian group with identity 1
2. V is closed under the scalar multiplication : ∀a∈V, ∀λ∈F: λa ∈V
3. There is distributivity for the scalar multiplication with respect to # in F:
∀a,b∈V, ∀λ∈F: λ(a#b) = (λa)#(λb)
4. There is distributivity for the scalar multiplication with respect to + in F:
∀a,b∈V, ∀λ∈F: λ(a+b) = (λa)+(λb)
5. Mixed associativity for • and *
∀a∈V, ∀λ∈F: λ(va) = (λ•v)a
6. identity of F{0} is also an identity for scalar multiplication:
∀a∈V: 1a = a

Question 32

vector space definition point 1

Answer 32

1. V,# is an abelian group with identity 1

Question 33

vector space definition point 2

Answer 33

2. V is closed under the scalar multiplication : ∀a∈V, ∀λ∈F: λa ∈V

Question 34

vector space definition point 3

Answer 34

3. There is distributivity for the scalar multiplication with respect to # in F:
   ∀a,b∈V, ∀λ∈F: λ(a#b) = (λa)#(λ*b)

Question 35

vector space definition point 4

Answer 35

4. There is distributivity for the scalar multiplication with respect to + in F:
   ∀a,b∈V, ∀λ∈F: λ(a+b) = (λa)+(λ*b)

Question 36

vector space definition point 5

Answer 36

5. Mixed associativity for • and *

∀a∈V, ∀λ∈F: λ(va) = (λ•v)*a

Question 37

vector space definition point 6

Answer 37

6. identity of F{0} is also an identity for scalar multiplication:
   ∀a∈V: 1*a = a

Question 38

subtraction binary operation

Answer 38

A subtraction is a binary operation on the vector space V defined as:
u-v = u + (-v)
∀u,v∈V (-v is the inverse of v)

Question 39

subspace

Answer 39

A non-empty subset U of a real vector space V is called a subspace under the same operations of addition and scalar multiplication relative to which V is a real vector space.

Question 40

linear combination

Answer 40

A vector v∈V, with a real vector space is a linear combination of the vectors u₁, u₂, … uₖ∈V if v can be written in the form:
v = λ₁u₁ + λ₂u₂ + … + λₖuₖ
where λ₁, λ₂,..,λₖ∈ℝ

Question 41

span

Answer 41

if u₁, u₂, … uₖ∈V, with V a real vector space. Then the subset of V consisting of all possible linear combinations of u₁, u₂, … uₖ is called their span.

Question 42

row space

Answer 42

Consider the matrix A∈Mₘₙ and let uᵢ denote the vector in ℝⁿ associated with the ith row in A. Then the row space of A is the subspace of ℝⁿ given by:
row(A) = span{ u₁, u₂, … uₘ}

Question 43

column space

Answer 43

Consider the matrix A∈Mₘₙ and let uᵢ denote the vector in ℝᵐ associated with the ith column in A. Then the coumn space of A is the subspace of ℝᵐ given by:
column(A) = span{ u₁, u₂, … uₙ}

Question 44

linearly independent

Answer 44

A set of vectors {u₁, u₂, … uₖ} in a real vector space V is said to be linearly independent when
λ₁u₁ + λ₂u₂ + … + λₖuₖ = 0
only when
λ₁ = λ₂ = … = λₖ = 0

Question 45

Linearly dependent

Answer 45

A set of vectors {u₁, u₂, … uₖ} in a real vector space V is said to be linearly dependent if there is at least one linear combination other than the trivial one which is equal to the zero vector:
λ₁u₁ + λ₂u₂ + … + λₖuₖ = 0
with at least one value of i= 1,2,…,k s.t. λᵢ!=0