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 ...

For a video explanation, click here 👉 https://youtu.be/3ByCs2ILhgo   Basis  :     A non-empty set S in a vector space V ( F ) is said to be a basis of  V if     i ) S is linearly independent   and    ii  ) S spans V i.e. L ( S ) = V .                                   Note  :          * In the above definition  of a Basis, the second condition L ( S ) = V means every vector in V is a          linear combination of vectors of S.      * For the two dimensional Euclidean Space  R 2 , for ( a , b ) in  R 2 ,we have                   ( a , b ) = a ( 1 , 0 ) + b ( 0 , 1 )               i.e. every element in  R 2 , is a linear combination of S = { ( 1 , 0 ) , ( 0 , 1 )...