devotion & mathematics

  For a video explanation, click here 👉 https://youtu.be/NIQCW-Jg9yo                               Cayley-Hamilton Theorem :  Statement :     Every square matrix satisfies its characteristic equation Proof :        * Basis Extension Theorem   * V/W forms a vector space  *  Show that the set { ( 1 , 2 , 1 ) , ( 2 , 1 , 0 ) , ( 1 , -1 , 2  } forms a basis of V3 ( F )   

For a video explanation, click the link 👉 https://youtu.be/unnIkQ7DhuM      THEOREM :      Let W be a subspace of a vector space V ( F ) . Then the set V/W = { W + 𝛂 / 𝛂 𝞊 V } of        all cosets forms a Vector Space under  the Coset addition and Scalar multiplication       are defined as        i.  For W + 𝛂 , W + 𝞫 𝞊 V,   ( W +  𝛂 ) + ( W +  𝞫 ) = W + ( 𝛂  +  𝞫 )       ii . For a 𝞊  F , W + 𝛂 𝞊 V,    a  ( W +  𝛂 ) =  ( W +  a𝛂 ).  PROOF :  * Find the coordinates of 𝜶 = (4 ,5 ,6) w.r.t.  x=(1,1,1),y = (-1,1,1),z = (1,0,-1)  *dim ( W1 + W2 ) = dim ( W1 ) + dim ( W2) - dim ( W1 ⋂ W2 )   *  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...

For a video explanation, click the link 👉 https://youtu.be/1_g20kjooQs  QUOTIENT SET :        Let W be a subspace of a vector space V ( F ) . Then the set V/W = { W + 𝛂 / 𝛂 𝞊 V } of        all cosets is called the Quotient Set in which the Coset addition and Scalar multiplication       are defined as        i.  For W + 𝛂 , W + 𝞫 𝞊 V,   ( W +  𝛂 ) + ( W +  𝞫 ) = W + ( 𝛂  +  𝞫 )       ii . For a 𝞊  F , W + 𝛂 𝞊 V,    a  ( W +  𝛂 ) =  ( W +  a𝛂 ). * Find the coordinates of 𝜶 = (4 ,5 ,6) w.r.t.  x=(1,1,1),y = (-1,1,1),z = (1,0,-1)  *dim ( W1 + W2 ) = dim ( W1 ) + dim ( W2) - dim ( W1 ⋂ W2 )   *  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 ...

For a video explanation , click here 👉  https://youtu.be/ydiEll71kLw Problem      Find the coordinates of 𝜶 = (4 ,5 ,6) w.r.t. x=(1,1,1),y = (-1,1,1),z = (1,0,-1). Solution : *  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                                  ...

For a video explanation , click here 👉 https://youtu.be/JQTyPoqUv6w?si=Njd0AGlMQ1AdpSUc Problem :      If 𝜶 = ( 1 , -1 , 0 ) , 𝜷 = ( 2 , 1 , 3 ) find a basis for R3 containing  𝜶 and 𝜷 . Solution :  * dim ( W1 + W2 ) = dim ( W1 ) + dim ( W2) - dim ( W1 ⋂ W2 ) *   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                                     ...

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 )...

For a video explanation, click here 👉 https://youtu.be/IyqGkHmIdKQ  Theorem :     If  𝜶,   𝜷, γ are linearly independent vectors in a vector space V    ( R ) then their pairwise sums  𝜶  +  𝜷        𝜷  + γ   ,  𝜶   +  γ  are also linearly independent.   Proof :             * 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                               ...