Pairwise sums of Linearly Independent Vectors : Linear Algebra : Degree

For a video explanation, click here 👉https://youtu.be/IyqGkHmIdKQ


 Theorem : 

   If 𝜶,  𝜷, γ are linearly independent vectors in a vector space V  ( R ) then their pairwise sums 𝜶 + 𝜷  

  𝜷 + γ   , 𝜶 + γ  are also linearly independent.

  Proof : 

         


 * Theorem on Linear Dependence 

* linear sum of subspaces

* What is a vector space

* linear span of a set 

  * L ( S ) = L ( S' )

 * 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

 * Basis and Dimension of a vector space





































































































#Pairwise #sums #of #Linearly #Independent #Vectors  : #Linear #Algebra  #: #Degree

#vector #algebra #vector algebra   #iit #jee #mains #dimension #basis #subspacec #linearalgebra                            

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