Problems on linear combination : Linear Algebra : Degree #Problems #on #linear #combination #Linear #Algebra #: #Degree

 Problems on linear combination : 

Problem 1 :       

          Express the vector 𝜶 = (1 , -2 , 5 ) as a linear combination of the vectors

e₁ = ( 1 , 1 , 1 ) , e₂ = ( 1 , 2 , 3 ) , e₃ = ( 2 , -1 , 1 ).

Solution :

Problem 2 :

Show that the vector 𝜶 = ( 2 , -5 , 3 ) in R³ can not be expressed as a

linear combination of the vectors e₁ = ( 1 , -3 , 2 ) ; e₂ = ( 2 , -4 , -1 ) ;

e₃ = ( 1 , -5 , 7 )

Solution :

* Linear combination of vectors

* linear sum of subspaces

* 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  #Linear #Algebra #: #Degree  

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