Remarks / Examples 1. If v 0, then {v} is a linearly independent set. Suppose cv = 0 for some c0. Then c-¹(cv) = 0. Usin

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Remarks Examples 1 If V 0 Then V Is A Linearly Independent Set Suppose Cv 0 For Some C0 Then C Cv 0 Usin 1 (109.11 KiB) Viewed 11 times

Remarks / Examples 1. If v 0, then {v} is a linearly independent set. Suppose cv = 0 for some c0. Then c-¹(cv) = 0. Using Property 7 of Definition 3.1, (c-¹c)v = 1v = 0 . Using Property 8 of Definition 3.1, this implies v = 0. 2. If one of the vectors V₁, V2, ···, vk is 0, then the set of vectors is linearly dependent. 3. A set of two vectors, {v₁, v2} is linearly dependent if and only if there exists a non-zero 4. If S is a linearly dependent set of vectors, and S ≤ T, then T is a linearly dependent set of vectors. Exercise 7. Prove items (2)-(4), as stated in the list given above. EF so that v₁ = av₂. To expand on the last of in this sequence of remarks, the intuition is that the more vectors you choose to include in a set, the more likely it is to span the vector space the vectors live in, but the less likely the vectors are to to be linearly independent. This observation leads to the idea of a basis. for a vector space.