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hw1sol - MATH 223 HOMEWORK#1 Solutions The problems in the...

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MATH 223 - HOMEWORK #1 Solutions The problems in the book define vector spaces over a field F . You may assume that we always have F = R . When solving the problems you may refer to any result we have proved in class or that is proved in the book. The numbers, such as 13 in Section 1.2, refer to the textbook. This first homework has the full text of the problems copied from the book since some students don’t have the textbook yet. Problem 1. Let V be a vector space. Use the axioms to prove: (i) For any vector v in V 0 · v = 0 . (Hint: add v + w = 0 to the left hand side and simplify, where w is the “negative” of v from axiom (VS 4).) (ii) For any vector v in V , the vector w = ( - 1) · v satisfies axiom (VS 4): v + w = 0. In other words, ( - 1) · v is the “negative” of v . (i) We can write: 0 · v = 0 · v + 0 = 0 · v + v + w. where we used axiom (VS 3) in the first equality and w comes from axiom (VS 4). Now v = 1 · v by axiom (VS 5)and we can continue: = 0 · v + 1 · v + w = (0 + 1) · v + w = v + w = 0 .
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