Linear sum is a subspace : Linear Algebra : Degree

For a video explanation, click here 👉https://youtu.be/DTwC9rO2rWQ

Theorem :   

         If  W1 and   W2 be two subspaces of the vector space V(F) . Then 

       1 )  W1 + W2   is a subspace of V(F)      and

       2 )  W1 ⊆  W1 + W2   and    W2   ⊆ W1 + W2   

 Proof : 


                                    


* Linear combination of vectors  

*  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










































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

#theorem #on #linear #algerbra   

                         


      

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