VGLA definitions term 2 Flashcards
Imaginary Unit
The Imaginary unit i is defined such that i²=-1 and (-i)²=-1
Complex numbers
The set of complex numbers ℂ is defined by ℂ = {a+bi: a∈ℝ and b∈ℝ}
equality of complex numbers
Two complex numbers (a+bi), (c+di) are equal if and only if
- a=c
- b=d
Sum of complex numbers
The sum of two complex numbers a+bi and c+di is the complex number (a+bi) + (c+ di) = (a+c) + (b+d)i
product of two complex numbers
The product of two complex numbers a+bi and c+di is the complex number (a+bi)(c+di) = (ac-bd)+(bc+ad)i
real part of a complex number
The real part of a complex number z=a+bi is given by Re(z) =a
Imaginary part of a complex number
The imaginary part of a complex number is given by Im(z)=b
complex conjugate
The complex conjugate of a complex number z=a+bi is given by z*=a+bi
modulus
The modulus of a complex number z=x+yi, |z| is given by |z|=√(a²+ b²)
Argument
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.
principle argument
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)
roots of complex numbers
a complex number ω is the nth root of a complex number z if and only if wⁿ=z
polynomial of degree n (real coefficiants)
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
zero
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
multiplicity
The number of times an item appears in an expression
|z-z₀| = α
The loci of points on a circle with centre z₀ and radius α
Arg(z- z₀) = θ
The loci of points on the half line starting at z₀ at the angle θ to the horizontal
|z-(a+bi)| = |z-(c+di)|
The loci of points on the perpendicular bisector between the points a+bi and c+di
Submatrix
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
(i,j)-minor
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
(i,j)-cofactor
The (i,j)-cofactor of a matrix A∈Mₙₙ, written as Cᵢⱼ(A), is defined by
Cᵢⱼ(A) = (-1)ᶦ⁺ʲMᵢⱼ(A)
determinant
The determinant of a matrix A∈Mₙₙ is given by det(A) = a₁₁C₁₁(A) + a₁₂C₁₂(A) + … + a₁ₙC₁ₙ(A)
Transpose Matrix
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
Cofactor Matrix
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)
Adjoint Matrix
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
binary operation
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)
internal binary operation
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.
group
A set V with a binary operation * is called a group under the binary operation * when the following conditions are satisfied:
- is an internal binary operation
- is associative
- has an identity
- for every element in V there exists an inverse element
Abelian group
A group under a binary operation * is abelian when * is commutative
field
A set V with binary operations * and # is called a field V,*,# when:
- V,* is an abelian group with identity 0
- V{0}, # is an abelian group with identity 1
- ∀x∈V: 0#x = x#0 = 0
- ∀x,y,z∈V: x#(yz) = (x#y)(x#z)
vectors space
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
vector space definition point 1
- V,# is an abelian group with identity 1
vector space definition point 2
- V is closed under the scalar multiplication : ∀a∈V, ∀λ∈F: λa ∈V
vector space definition point 3
- There is distributivity for the scalar multiplication with respect to # in F:
∀a,b∈V, ∀λ∈F: λ(a#b) = (λa)#(λ*b)
vector space definition point 4
- There is distributivity for the scalar multiplication with respect to + in F:
∀a,b∈V, ∀λ∈F: λ(a+b) = (λa)+(λ*b)
vector space definition point 5
- Mixed associativity for • and *
∀a∈V, ∀λ∈F: λ(va) = (λ•v)*a
vector space definition point 6
- identity of F{0} is also an identity for scalar multiplication:
∀a∈V: 1*a = a
subtraction binary operation
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)
subspace
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.
linear combination
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
span
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.
row space
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}
column space
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}
linearly independent
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
Linearly dependent
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
