SUBMITTED BY: Ms. Harjeet Kaur Associate Professor Department of Mathematics PGGCG – 11, Chandigarh DEFINITION: LINEAR COMBINATION Let V(F) be a vector space. A vector v ∈ V(F) is said to be linear combination of the vectors 푣1, 푣2,……, 푣푛 ∈ V(F), provided ∃ scalars ∝1, ∝2,……………, ∝푛 ∈ F such that v = ∝1 푣1+ ∝2 푣2 + … . + ∝푛 푣푛 푛 = 푖=0 ∝푖 푣푖

DEFINITION: LINEAR SPAN Let V(F) be a vector space and S be a non-empty subset of V. The linear span of S is the set of all linear combinations of any number of elements of S and is denoted by S.

푛 Thus, L(S) = 푖=1 ∝푖 푣푖 : 푣푖 ∈ S, ∝푖 ∈ F, 1 ≤ i ≤ n

DEFINITION: Generator of Vector Space

Let V(F) be a vector space. A non-empty subset S of V(F) is called a generator of V(F) if every element of V can be expressed as the linear combination of elements of S.

Theorem: Prove that the linear span L(S) of any subset S of vector space V(F) is a sub-space of V generated by S. Moreover L(S) = {S} THEOREM: If S and T are any two subsets of a vector space V(F), prove that:

(i) S ⊂ L(T) ⟹ L(S) ⊂ L(T) (ii) S ⊂ T ⟹ L(S) ⊂ L(T) (iii) L(S ∪ T) = L(S) + L(T) (iv) S is a subspace of V iff L(S) = S (v) L(L(S)) = L(S) Example: Express the vector v = (1,-2,5) as the linear combination of vectors 푣1 = (1,1,1), 푣2 = (1,2,3) and 3 푣3 = (2,-1,1) in 푅 (R). v = -6 푣1 + 3 푣2 + 2 푣3 3 1 Example: The matrix M= is 1 −1 linear combination of the matrices 1 1 0 0 0 2 A= , 퐵 = ,C= 1 0 1 1 0 −1 as M=3A-2B-C 3 Example: In a vector space 푅 let 푣1 = (1,2,1), 푣2 = (3,1,5) and 푣3 = (3,-4,7). Prove that the subspaces spanned by S = {푣1, 푣2} and T = {푣1, 푣2, 푣3} are the same. DEFINITION: Linear Dependence

Let V(F) be a vector space. A finite set {푣1, 푣2,…….., 푣푛} of vectors V is said to be linearly dependent if there exist scalars ∝1, ∝2,……………, ∝푛 ∈ F, not all zero, such that

∝1푣1+ ∝2 푣2 + … + ∝푛 푣푛 = 0

DEFINITION: Linear Independence Let V(F) be a vector space. A finite set {푣1, 푣2,…….., 푣푛} of vectors V is said to be linearly independent if every relation of the form ∝1푣1+ ∝2 푣2 + … + ∝푛 푣푛 = 0, ∝푖’s ∈ F, 1 ≤ i ≤ n ⇒ ∝푖 = 0, ∀ 1 ≤ i ≤ n

Note: Any infinite set of vectors of V is said to be linearly independent if its every finite subset is linearly independent. THEOREM: Let V be a vector space over a field F. Prove that

(i) Every non-zero singleton subset of V is linearly independent over F, (ii) Every set containing only zero vector is linearly dependent over F

THEOREM: Let V be a vector space over a field F. Prove that

(i) Every super set of a L.D. set of vector L.D. over F (ii) Any subset of a L.I. set of vectors is L.I.

THEOREM: Let V be a vector space over a field F. Prove that

(i) The set {푣1, 푣2} is L.D. iff 푣1 and 푣2 are collinear (ii) The set {푣1, 푣2, 푣3} is L.D. iff 푣1, 푣2 and 푣3 are coplanar

THEOREM: Let V be a vector space over the field F. Prove that the set of non-zero vectors 푣1, 푣2,……, 푣푛 ∈ V is L.D. iff some of these vectors, say 푣푘, 2 ≤ k ≤ n, can be expressed as the linear combination of the preceeding vectors of set S. EXAMPLE: Show that the vectors 푣1 = (1,2,0); 푣2 = (0,3,1) and 푣3 = (-1,0,1) in 푉3(R) are L.I.

∝(1,2,0) + β(0,3,1) + γ(-1,0,1) = (0,0,0) (∝- γ, 2 ∝ + 3 β, β+ γ) = (0,0,0) ∝- γ = 0 2 ∝ + 3 β = 0 β+ γ = 0 i.e. ∝ = β = γ = 0