Linear Dependence and Independence

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Theorem: Let V be a vector space, and let S_1S_2V. If S_1 is linearly dependent, then S_2 is linearly dependent.
Corollary: Let V be a vector space, and let S_1S_2V. If S_2 is linearly independent, then S_1 is linearly independent.
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Theorem: If given S_1 ⊆ S_2 ⊆ V: S_1 is linearly dependent then S_2 is linearly dependent we begin by assuming S_1 is linearly dependent. Then for some arbitrary set of vectors  u_1,...,u_n in S_2, then for all a_1,...,a_n in R, with at least some a_i ≠ 0, 1 ≤ i ≤ n:

0 = a_1*u_1 + ... + a_n*u_n.

Since S_1 ⊆ S_2 then u_1,...,u_n ∈ S_2, hence S_2 must be linearly dependent.
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Corollary: In order to prove that given S_1 ⊆ S_2 ⊆ V: S_2 is linearly independent then S_1 is also linearly independent one first assumes that S_2 is linearly independent. If so, then that means that the arbitrary set of vectors u_1,...,u_n in S_2 are all unique (ie: ∃! u_1,...,u_n ∈ S_2). Therefore: if S_2 ⊇ S_1 then all of the vectors of S_1 are also unique. Hence this proves the corollary. []

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