Apprentice Program: Linear Algebra
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چکیده
Definition 1.1. Let K be a field. (V,+, ·) is a vector space over K, if (V,+) is an Abelian group, and · : K × V → V (called scalar multiplication) is distributive, associative and multiplication by 1 is the identity on V . This compact definition unwinds to give us the following ten(!) axioms: (1) + : V × V → V (2) u+ (v + w) = (u+ v) + w. (3) u+ v = v + u. (4) ∃0 ∈ V such that v + 0 = 0 + v = v, for all v ∈ V . (5) ∀v ∈ V , ∃ − v ∈ V such that v + (−v) = 0. (6) · : K × V → V . (7) (αβ) · v = α · (β · v). (8) (α+ β) · v = α · v + β · v. (9) α · (u+ v) = α · u+ α · v. (10) 1 · v = v.
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