Theorem on Linear Dependence : Linear Algebra : Degree #Theorem #on #Linear #Dependence #: #Linear #Algebra #Degree
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 : * 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 ?...