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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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L ( S ) = L ( S' ) : Linear Algebra : Degree #L(S)=L(S') : #Linear #Algebra #: #Degree

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For a video explanation, click here 👉 https://youtu.be/r3u7AVlNg-Y Theorem : Let S = { 𝜶1 , 𝜶2 , ... , 𝜶n } be a subset of the vector space V ( F ) . If 𝜶i in S is a linear combination of its preceeding vectors then L ( S ) = L ( S' ) where S' = { 𝜶1 , 𝜶2 , ... , 𝜶(i-1) , 𝜶(i+1) , ... , 𝜶n } . Proof :  * Theorem on Linear Dependence  * linear sum of subspaces * What is a vector space * linear span of a set    * 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 ...
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Theorem on Linear Dependence : Linear Algebra : Degree #Theorem #on #Linear #Dependence #: #Linear #Algebra #Degree

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For a video explanation , click here 👉   https://youtu.be/8T7JZOyRO70   Theorem :       Let V ( F ) be a vector space and S = { 𝜶 1 , 𝜶 2 , … , 𝜶 n } is a finite subset of non – zero vectors of V ( F ). Then S is linearly dependent if and only if some vector 𝜶 k 𝟄 S , 2 ≤ k ≤ n , can be expressed as a linear combination of its preceeding vectors. Proof :   * L ( S ) = L ( S' ) * linear sum of subspaces * What is a vector space * linear span of a set    * 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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L(S) is a subspace : Linear Algebra : Degree #L(S) #is #a #subspace #Linear #Algebra #: #Degree

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For a video explanation, click the link 👉 https://youtu.be/NuvBv7ppF2k Theorem :         For any subset S of a vector space V ( F ) , the linear span of S L ( S ) is a subspace of V.   Proof :                           * Linear combination of vectors * linear sum of subspaces * What is a vector space  * 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 suf...
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L ( W1 ⋃ W2 ) = W1 + W2 : Linear Algebra: Degree #L(W1⋃W2) #= #W1 #+ #W2 : #Linear #Algebra #: #Degree

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For a video explanation , click here 👉 https://youtu.be/DvN0WBjcOjY Theorem :      If W 1 and W 2 are any two subspace of a vector space V ( F ) then            L ( W 1 ⋃ W 2  ) = W 1 + W 2 . Proof :          * Linear combination of vectors * linear sum of subspaces * What is a vector space  * 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  ...
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Problems on linear combination : Linear Algebra : Degree #Problems #on #linear #combination #Linear #Algebra #: #Degree

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For a video explanation, click here 👉 https://youtu.be/BdD_ET3MfoA   Problems on linear combination :  Problem 1 :                   Express the vector 𝜶 = (1 , -2 , 5 ) as a linear combination of the vectors e 1 = ( 1 , 1 , 1 ) , e 2 = ( 1 , 2 , 3 ) , e 3 = ( 2 , -1 , 1 ). Solution : Problem 2 : Show that the vector 𝜶 = ( 2 , -5 , 3 ) in R 3 can not be expressed as a linear combination of the vectors e 1 = ( 1 , -3 , 2 ) ; e 2 = ( 2 , -4 , -1 ) ; e 3 = ( 1 , -5 , 7 ) Solution : * Linear combination of vectors * linear sum of subspaces * What is a vector space  * 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,...
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