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Basis Extension Theorem : Linear Algebra : Degree #Basis #Extension #Theorem #: #Linear #Algebra #: #Degree

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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 ...
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dim ( W1 + W2 ) = dim ( W1 ) + dim ( W2) - dim ( W1 ⋂ W2 ) : Linear Algebra : Degree #most #important #theorem #linear #algebra

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                                            [  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 )...
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Find the coordinates of 𝜶 = (4 ,5 ,6) w.r.t. x=(1,1,1),y = (-1,1,1),z = (1,0,-1) :linear algebra : Degree

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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                                  ...
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If 𝜶 = ( 1 , -1 , 0 ) , 𝜷 = ( 2 , 1 , 3 ) find a basis for R3 containing 𝜶 and 𝜷 : Linear Algebra : Degree #Linear #Algebra #: #Degree

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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                                     ...
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Show that the set { ( 1 , 2 , 1 ) , ( 2 , 1 , 0 ) , ( 1 , -1 , 2 } forms a basis of V3 ( F ) : Linear Algebra : Degree #linear #algebra #degree

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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 ...
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Basis & Dimension : Linear Algebra : Degree #Basis #& #Dimension #: #Linear #Algebra #: #Degree

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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 )...
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Pairwise sums of Linearly Independent Vectors : Linear Algebra : Degree

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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                               ...
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