dim ( W1 + W2 ) = dim ( W1 ) + dim ( W2) - dim ( W1 ⋂ W2 ) : Linear Algebra : Degree #most #important #theorem #linear #algebra

                                            [  Most  Important Theorem  ]

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Theorem :

If W1 and W2 be any two subspaces of a finite dimensional vector space V ( F )

then dim ( W1 + W2 ) = dim ( W1 ) + dim ( W2) - dim ( W1 ⋂ W2 )

 Proof : 

    

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

*  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 

#most #important #theorem #google #repeated 

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

#If #W1 #and #W2 #be #any #two #subspaces #of #a #finite #dimensional #vector #space 

#V(F)                                               

#then #dim(W1+W2) #= #dim(W1) #+ #dim(W2) #- #dim(W1⋂W2)