Linear Sum of Subspaces : Linear Algebra : Degree

Linear Sum of Subspaces: Definition 
                

             Let   W1 and   W2 be two subspaces of the vector space V(F) . Then the linear sum of the subspaces W1 & W2 , denoted by W1 + W2, is the set of all sums 𝜶1+ 𝜶2 such that  𝜶1 𝞊 W1,  𝜶2 𝞊 W2 i.e. W1 + W2 = { 𝜶1+ 𝜶2 / 𝜶1 𝞊 W1,  𝜶2 𝞊 W2 }.  



* 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










































































































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

#definition #linearsum #of   #linear #sum                  


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