devotion & mathematics

                                            [  Most  Important Theorem  ] For a video explanation, click here 👉 https://youtu.be/zlzgpCKztdI 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' )  * Theore...

  CONTENTS : Theorem  * : Cantor's  Intersection Theorem Theorem ** :  Let {x n } be a sequence in a metric space (X,d). For each positive integer n, write  E  n ={x  m /n ≥ m}. Then { x  n } is a Cauchy sequence in X if and only if lim diamE n =0. Theorem *** :  For any subset E of a metric space (X,d), diam E=diam Ē. The following are the proofs of the above statements : proofs : Theorem * : Cantor Intersection Theorem  Statement & Proof :     Theorem **  :  Let {x n } be a sequence in a metric space (X,d). For each positive integer n, write  E  n ={x  m /n ≥ m}. Then { x  n } is a  Cauchy sequence  in X if and only if lim diamE n =0. Proof :   Theorem ***  :  For any subset E of a metric space (X,d), diam E=diam Ē. Proof :                    people also ask :  1. What is a metric space?  2. What i...