devotion & mathematics

For a video explanation, click here 👉 https://youtu.be/qsG7n-7nhx4   Problem : Show that the set { ( 1 , 2 , 1 ) , ( 2 , 1 , 0 ) , ( 1 , -1 , 2 } forms a basis of V3 ( F ). Solution :  * 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 ...

 Repeated Integral   :                                  Let f(x) be a bounded function defined on the rectangle R=[a,b;c,d].            Suppose  ∫ f(x ,y) dy exists for each x  Ꜫ [a ,b] and y limits from c to d. If g(x) = ∫ f( x ,y )dy, y limits from c to d , then g(x) is a function on [a ,b] . If g(x) is also integrable on [a ,b] , then ∫g(x)dx =  ∫[  ∫ f(x , y)dy ] dx is called a repeated integral of f(x ,y) on R where x limits from a to b and y limits from c to d.                                                            Problems on Repeated Integrals : 1. Evaluate  ∫[  ∫ (x+y+1) dx ] dy and  ∫[ ∫(x+y+1) dy]dx , x limits from -1 to 1 and y limits are from -1 to 0. 2. Show tha...