Linear Span of a set : Linear Algebra : Degree #linear #span #of #a #set

 Linear Span of a set :          

                                       Let S be a non-empty subset of a vector space V (F). Then the set of 

all linear combinations of vectors of S is called the linear span of S and is denoted by L (S).

         i.e. L (S) = { 𝜶 / 𝜶 = 𝜮 , 1 ≤ i ≤ n } .

For example : 

          If S = { 𝜶₁, 𝜶₂, … , 𝜶_n .} and if 𝜶 𝞊 L (S) then there exist scalars a₁, a₂, … , a_n  such            that  𝜶 = a₁ 𝜶₁ +a₂ 𝜶₂ +  … +a_n 𝜶_n .

* Linear combination of vectors

* linear sum of subspaces

* What is a vector space

 * Theorem on vector space  

 * Historical Introduction to Linear Algebra

 * Let V(F) be a vector space and let W ⊆ V. The necessary and sufficient conditions for W 

            to be a subspace of V are (i) 𝜶 𝞊 W, 𝞫 𝞊 W ⇒  𝜶 - 𝞫 𝞊 W

                                                     (ii) a 𝞊 F , 𝜶 𝞊 W ⇒ a𝜶 𝞊 W.

* Let V(F) be a vector space and let W ⊆ V. The necessary and sufficient conditions for W 

    to be a subspace of V is  𝜶 𝞊 W, 𝞫 𝞊 W ⇒  a𝜶 +b𝞫 𝞊 W

 * Problem 1 on subspace of a vector space 

* Problem 2 on subspace of a vector space

 * Intersection of subspaces is again a subspace

  * union of subspaces of a vector space

 * Problems on linear combination

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 #linear #span #of #a #set