Find the coordinates of 𝜶 = (4 ,5 ,6) w.r.t. x=(1,1,1),y = (-1,1,1),z = (1,0,-1) :linear algebra : Degree

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Problem 

   Find the coordinates of 𝜶 = (4 ,5 ,6) w.r.t. x=(1,1,1),y = (-1,1,1),z = (1,0,-1).

Solution :

*  If 𝜶 = ( 1 , -1 , 0 ) , 𝜷 = ( 2 , 1 , 3 ) find a basis for R3 containing 𝜶 and 𝜷 

*  Show that the set { ( 1 , 2 , 1 ) , ( 2 , 1 , 0 ) , ( 1 , -1 , 2  } forms a basis of V3 ( F )

 * Theorem on Linear Dependence 

* linear sum of subspaces

* What is a vector space

* linear span of a set 

  * L ( S ) = L ( S' )

 * 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 sums of Linearly Independent Vectors 

#Find #the #coordinates #of #𝜶 = #(4 ,5 ,6) #w.r.t. #x= #(1,1,1), #y = #(-1,1,1) #, #z #= #(1,0,-1)