Vectors, Spaces Flashcards
Linear combination of two vectors
X,Y = aX+bY
where a,b∈R, X,Y∈V (here V is set of vectors)
Explain the multiplication of vector with scalar
If α∈R then αX = α[x₁ x₂ ….Xₙ]
What is linear combination of one vector
Multiplication of vector with scalar
If α∈R then αX = α[x₁ x₂ ….Xₙ]
What is the scalar multiple of vector
Linear combination of one vector
If α∈R then αX = α[x₁ x₂ ….Xₙ]
Operations of vectors
Addition
Subtraction
Multiplication of vector with scalar
Products of vectors - 1.Dot Product 2.Cross Product
How many vectors generate by linear combination of vectors
Infinite number of vectors(space)
Type of product of vectors
Two type
1.)Dot Product
2.)Cross product
Another name of dot product
Inner product
Give me two another names of dot product
Inner product
Scalar product
Dot product gives what
Scalar quantity
Notation for dot product
All three notations for dot product
Give mathematical formula of dot product
<X,Y> = XᵀY
Is <X,Y> = <Y,X>
Yes
<X,Y> = <Y,X> it implies what
XᵀY = YᵀX
If dot product is zero then
vectors are orthogonal vectors
i.e angle between vectors is 90degree
If vectors are orthogonal then
their dot product is zero
Norm of vector
Notation, definition, formula
Length of vector
Normalised vector
(another name, definition, formula, notation of all types)
Orthonormal vectors
Unit vector of x, y and z respectively
i cap, j cap, k cap
orthonormal system (definition another name)
The space generated by orthonormal vectors called orthonormal system or orthonormal space
what type of i cap, j cap, k cap vectors are
Orthonormal vectors
i.e. orthogonal and unit vectors
Rⁿ
Rⁿ is n dimensional real space of vectors
Special note about orthogonal vectors
All orthogonal vectors are linearly independent vectors but all independent vectors are need not to be orthogonal.
Linear span
Let S = {V₁, V₂, V₃, …..Vₖ} is non empty subset of a vector space ‘V’ . (S⊆V)
Set of all linear combinations of vectors in ‘S’ is called Linear span.
How to write linear span of X,Y vectors
LS = {aX + bY | a, b ∈ R and X, Y ∈ S}
Linear Dependency (definition)
A set of vectors {X₁, X₂, X₃, …… Xₙ} is linearly dependent if their linear combination a₁X₁, a₂X₂, a₃X₃, …… aₙXₙ = 0 for not all (some) aᵢ = 0 ∈ R
Linearly independent (definition)
A set of vectors {X₁, X₂, X₃, …… Xₙ} is linearly dependent if their linear combination a₁X₁, a₂X₂, a₃X₃, …… aₙXₙ = 0 only for all aᵢ = 0
How can we determined that vectors are linearly dependent or independent
If any vector can be expressed as linear combination of other vectors in set then they are linearly dependent.
When can we say that vector is spanned
If any vector is obtained by linear combination of some vectors, we call it as spanned.
Does division and multiplication comes under linear combination
No
How can we say that two vectors are independent or not
if we have two vectors in the form
V1 = (a1 b1 c1) and V2 = (a2 b2 c2)
then if a1/a2 = b1/b2 = c1/c2 then they are linearly dependent
Vector space
Vector Space
A vector space (also known as a linear space) is a collection of objects called vectors, which can be added together and multiplied by numbers (scalars). The operations of vector addition and scalar multiplication in a vector space must satisfy certain properties or axioms.
Key Properties of a Vector Space:
- Vector Addition: If u and v are vectors in the vector space V,then their sum u + v is also in V.
- Scalar Multiplication: If v is a vector in V and c is a scalar, then the product cv is also in V.
- Commutativity: u + v = v + u for all vectors u, v in V.
- Associativity: (u + v) + w = u + (v + w) for all vectors u, v, w in V.
- Additive Identity: There exists a zero vector 0 in V such that v + 0 = v for all vectors v in V.
- Additive Inverse: For each vector v in V, there exists a vector -v such that v + (-v) = 0.
- Distributivity of Scalar Multiplication with Respect to Vector Addition: c(u + v) = cu + cv for all vectors u, v in V and scalar c.
- Distributivity of Scalar Multiplication with Respect to Scalar Addition: (c + d)v = cv + dv for all vectors v in V and scalars c, d.
- Associativity of Scalar Multiplication: c(dv) = (cd)v for all vectors v in V and scalars c, d.
- Multiplicative Identity: 1v = v for all vectors v in V, where 1 is the multiplicative identity in the field of scalars.
Subspace
Subspace
A subspace is a subset of a vector space that is itself a vector space under the same operations of vector addition and scalar multiplication as the original vector space.
Criteria for a Subspace:
For a subset W of a vector space V to be considered a subspace, it must satisfy the following three conditions:
- Non-Empty (Contains the Zero Vector): The zero vector 0 of V must be in W. This ensures that W is non-empty.
- Closed Under Addition: If u and v are vectors in W, then their sum u + v must also be in W.
- Closed Under Scalar Multiplication: If v is a vector in W and c is a scalar, then the product c*v must be in W.
Relationship between linear span, vector space, subspace
L(S) or you can say linear span of any non-empty subset ‘S’ of vector space ‘V’ is a subspace of V.
Basis of a vector space
A non empty subset of a vector space ‘V’ is said to be basis of ‘V’ if
1.) All the vectors of ‘S’ are linearly independent
2.) L(S) = V
we can write second point as:
the entire V is generated by the LC of vectors of S
or
Linear span of all the vectors S generates the entire vector space
or
Linear combination of vectors in ‘S’ generates all the vectors of V
Note on basis
Basis of any vector space is not unique.
Dimension of vector space
Dimension of vector space os equal to number of independent vectors in it
Relationship between Dimension and basis
Dimension(V) = number of elements in basis
[Here V is vector space]
Dimension of Rⁿ
Dimension of Rⁿ = n
If W₁ and W₂ are Subspaces of Vector space ‘V’ then
_______________
________________
(2 points)
1.) W₁+W₂ is also subspace of ‘V’
2.) dim(W₁U W₂) = dim(W₁) + dim(W₂) - dim(W₁∩ W₂)
Inequalities of dot product (2)
How to find angle between two vectors
