hw1 - MATH 223 - HOMEWORK #1 Due Friday, September 14 The...

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Unformatted text preview: MATH 223 - HOMEWORK #1 Due Friday, September 14 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 . Problem 2. 13 in Section 1.2. Let V denote the set of ordered pairs of real numbers. If (a1 , a2 ) and (b1 , b2 ) are elements of V and c ∈ R, define (a1 , a2 ) + (b1 , b2 ) = (a1 + b1 , a2 b2 ), c(a1 , a2 ) = (ca1 , a2 ). Is V a vector space with these operations? Justify your answer. Problem 3. 21 in Section 1.2. Let V and W be vector spaces. Let Z = {(v, w)|v ∈ V, w ∈ W }. Prove that Z is a vector space with the operations (v1 , w1 ) + (v2 , w2 ) = (v1 + v2 , w1 + w2 ), c(v1 , w1 ) = (cv1 , cw1 ). Please check axioms 1-4 only. This space Z is known as the product V × W . Problem 4. Let V be the set of positive real numbers, V = {a ∈ R|a > 0}. We write a vector in V with an arrow to distinguish it from a scalar; for example, a, 3, π are all elements of V . Define addition and scalar product as follows: → − −→ − a + b = a · b, c · a = ac . Is V with these operations a vector space? Justify your answer. (You have to be careful here with the zero vector. In checking (VS 3) you have to determine which vector is the zero vector, and then use the same zero vector in (VS 4).) 1 ...
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This note was uploaded on 02/01/2010 for the course MATH math115a taught by Professor Shalom during the Spring '10 term at UCLA.

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