Theorem on Linear Dependence : Linear Algebra : Degree #Theorem #on #Linear #Dependence #: #Linear #Algebra #Degree

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Theorem : 

   Let V ( F ) be a vector space and S = { 𝜶₁, 𝜶₂, … , 𝜶_n } is a finite subset of non – zero vectors of V ( F ). Then S is linearly dependent if and only if some vector 𝜶_k 𝟄 S , 2 ≤ k ≤ n , can be expressed as a linear combination of its preceeding vectors.

Proof : 

* 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

 #Theorem #on #Linear #Dependence #: #Linear #Algebra    #Degree

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

#Let V ( F ) #be #a #vector #space #and #S v= #{ 𝜶₁, 𝜶₂, … , 𝜶_n } #is a #finite #subset #of #non #– #zero vectors #of V ( F ). #Then S is #linearly #dependent #if #and #only #if #some #vector #𝜶_k 𝟄 #S #, #2 #≤ k ≤ n , #can #be #expressed #as a linear combination of #its #preceeding vectors.