Viewing posts tagged n x n

Theorem: Let **V** be a **vector space**, and let **S_1** ⊆ **S_2** ⊆ **V**. If **S_1** is **linearly dependent**, then **S_2** is** linearly dependent**.

Corollary: Let **V** be a **vector space**, and let **S_1** ⊆ **S_2** ⊆ **V**. 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:

Theorem: *A subspace*

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Let W_n be the subset of all M_(n x n)(

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