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How to count from one to one quintillion

Linear Dependence and Independence

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:

Symmetric Matrices as a Subspace of all Square Matrices

Theorem: subspace W of a vector space V over R is a subset of V which also has the properties that W is closed under addition and scalar multiplication. That is, For all x, y in W, x and y are in V and for any c in R, cx + y is in W.
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Let W_n be the subset of all M_(n x n)(R) such that A_ij = A_ji (ie. the set of all real symmetric square matrices).