devotion & mathematics

For a video explanation, click here 👉 https://youtu.be/b0KvzStFHAc   Linearly Independent Vectors :          Let 𝜶 1 , 𝜶 2 , … , 𝜶 n be any n vectors in a vector space V ( F ). Then the      vectors 𝜶 1 , 𝜶 2 ,  … , 𝜶 n are   called Linearly Independent vectors , if there exists       Scalars a 1 , a 2 , … , a n   such that  a 1 𝜶 1 + a 2 𝜶 2 + … + a n 𝜶 n = 0 implies a 1 = a 2 = … = a n = 0.      Linearly Dependent Vectors :          Let 𝜶 1 , 𝜶 2 , … , 𝜶 n be any n vectors in a vector space V ( F ). Then the      vectors 𝜶 1 , 𝜶 2 , … , 𝜶 n are   called Linearly Dependent vectors , if there exists scalars      a 1 , a 2 , … , a n  not all zero  such that   a 1 𝜶 1 + a 2 𝜶 2 + … + a n 𝜶 n = 0 

       Theorem :      A singleton set with non-zero vector is Linearly Independent   Proof :           Suppose V ( F) is a vector space and S = {𝜶} is a subset of V where 𝜶 ≠ 0 .         Now we prove S is Linearly Independent.          For a 𝞊 F such that a𝜶 = 0                                      ⇒ a=0 or 𝜶 = 0  .          Since 𝜶 ≠ 0 , we have a = 0.            ∴ S is Linearly Independent.                                               ***Hence the proof ***      Explanation :           Linearly independent set means if the linear combination of vectors is ...

  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 the link 👉   https://youtu.be/nFliuHhXhRc?si=_Ktb9UG9aDTYf6Lw   Basis Extension Theorem : Statement :              Let V ( F ) be a finite dimensional vector space and  S  = { 𝜶 1 , 𝜶 2 , … , 𝜶 n } a linearly independent subset of V.  Then S is itself a basis or it can be extended to form a basis of V.  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  } 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 ...

                                            [  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  * Basis Extension Theorem * linear sum of subspaces * What is a vector space * linear span of a set      * L ( S )...

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                                     ...