Vectors, Spaces Flashcards

Question

Special note about orthogonal vectors

Answer 1

All orthogonal vectors are linearly independent vectors but all independent vectors are need not to be orthogonal.

Answer 2

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.

Answer 3

LS = {aX + bY | a, b ∈ R and X, Y ∈ S}

Answer 4

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

Answer 5

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

Answer 6

If any vector can be expressed as linear combination of other vectors in set then they are linearly dependent.

Answer 7

If any vector is obtained by linear combination of some vectors, we call it as spanned.

Answer 8

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

Answer 9

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: 1. Vector Addition: If u and v are vectors in the vector space V,then their sum u + v is also in V. 2. Scalar Multiplication: If v is a vector in V and c is a scalar, then the product cv is also in V. 3. Commutativity: u + v = v + u for all vectors u, v in V. 4. Associativity: (u + v) + w = u + (v + w) for all vectors u, v, w in V. 5. Additive Identity: There exists a zero vector 0 in V such that v + 0 = v for all vectors v in V. 6. Additive Inverse: For each vector v in V, there exists a vector -v such that v + (-v) = 0. 7. Distributivity of Scalar Multiplication with Respect to Vector Addition: c(u + v) = cu + cv for all vectors u, v in V and scalar c. 8. Distributivity of Scalar Multiplication with Respect to Scalar Addition: (c + d)v = cv + dv for all vectors v in V and scalars c, d. 9. Associativity of Scalar Multiplication: c(dv) = (cd)v for all vectors v in V and scalars c, d. 10. Multiplicative Identity: 1v = v for all vectors v in V, where 1 is the multiplicative identity in the field of scalars.

Answer 10

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: 1. Non-Empty (Contains the Zero Vector): The zero vector 0 of V must be in W. This ensures that W is non-empty. 2. Closed Under Addition: If u and v are vectors in W, then their sum u + v must also be in W. 3. 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.

Answer 11

L(S) or you can say linear span of any non-empty subset 'S' of vector space 'V' is a subspace of V.

Answer 12

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

Answer 13

Basis of any vector space is not unique.

Answer 14

Dimension of vector space os equal to number of independent vectors in it

Answer 15

Dimension(V) = number of elements in basis [Here V is vector space]

Answer 16

Dimension of Rⁿ = n

Answer 17

1.) W₁+W₂ is also subspace of 'V' 2.) dim(W₁U W₂) = dim(W₁) + dim(W₂) - dim(W₁∩ W₂)