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 )...

  TETRAHEDRON : A tetrahedron is a third dimensional structure having 4 triangles as faces, six edges and four vertices.  A tetrahedron is also called a triangular pyramid.   In a tetrahedron if all four triangles are equilateral triangles then the tetrahedron is called a regular tetrahedron. The volume of a tetrahedron is one-third the area of the base times the height i.e. volume V= (1/3)bh, where b is base and h is height. Also the total surface area is the sum of the surface areas of all it's faces. Here We see some properties of a regular tetrahedron.  Volume = a^3/6√2 Total surface area= (a^2)√3 surface area of each face = (1/4)(a^2)√3 Height =a√6/3,                             here a is the side of regular tetrahedron. #tetrahedron #volume #surfacearea #height #dimension #geometry

FEW FOURTH DIMENSIONAL OBJECTS   HYPERCUBE :                          A hypercube is 4th dimensional analog of a cube . Hypercube is essentially a cube extended into a fourth  dimension.    A hypercube has 8  cubes(3D) as faces, 16 vertices(0 D), 32 edges(1D),24 faces  which are 2D squares.                     { Here D indicates the dimension}   In this hypercube all edges meet at right angles.          Basically if we take a 3D cube , a 3D cube has height, width, breath and area whereas the 4D cube i.e. a hypercube has all the above height, width, breath and area thrice i.e. inner, outer and total.  That means hypercube has inner height, outer height and total height,  inner width, outer width and total width just like the  other two breath and area are also measured for inner, outer and total ...