Necessary Condition 3 for subspace : LInear Algebra : Degree

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Theorem : 

                  A non-empty set W is a subset  of a vector space V(F)  . W is a subspace of V

                 if and only if a 𝞊 F and 𝝰 , 𝞫 𝞊 W ⇒ a𝝰 + 𝞫 𝞊 W .

 Proof:

          

* 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

 

           

 #A #non #- #empty #set #W #is #a #subset  #of #a #vector #space #V(F)  . #W #is #a #subspace #of #V

                 #if #and #only #if #a #𝞊 #F #and #𝝰 #, #𝞫 #𝞊 #W #⇒ #a𝝰 #+ #𝞫 #𝞊 #W #.

 #Anon-emptysetWisasubsetofavectorspaceV(F).WisasubspaceofVifandonlyifa𝞊Fand𝝰,𝞫𝞊W⇒a𝝰+𝞫𝞊W .

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