L ( S ) = L ( S' ) : Linear Algebra : Degree #L(S)=L(S') : #Linear #Algebra #: #Degree

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Theorem :

Let S = { 𝜶1 , 𝜶2 , ... , 𝜶n } be a subset of the vector space V ( F ) . If 𝜶i in S is a linear combination of its preceeding vectors then L ( S ) = L ( S' ) where S' = { 𝜶1 , 𝜶2 , ... , 𝜶(i-1) , 𝜶(i+1) , ... , 𝜶n } .

Proof :

 * Theorem on Linear Dependence 

* linear sum of subspaces

* What is a vector space

* linear span of a set 

 * 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

 * Pairwise sum of linearly independent sets 

#L(S)=L(S')  :  #Linear #Algebra   #: #Degree 

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