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Unformatted text preview: Math 115A Homework # 1 Solutions Prof. Yehuda Shalom TA: Darren Creutz Date Due: 14 Jan 2010 Section 1.2 # 1 a) T: VS axiom 3 b) F: Lemma proved in discussion (using Cancellation) c) F: when x = 0 we get that 0 x = 1 x but 0 6 = 1 d) F: when a = 0 we get that 0 x = 0 y for every x and y e) T: Example 1 f) F: is has m rows and n columns g) F: P ( F ) is a vector space so any two polynomials can be added h) F: consider x and ( 1) x which are both degree one, but x + ( 1) x = 0 is of degree 0 i) T: definition of scalar multiplication for polynomials j) T: definition of polynomials k) T: definition of functions Section 1.2 # 21 Let V and W be vector spaces over the same field F . Let Z = { ( v, w ) : v ∈ V, w ∈ W } Prove that Z is a vector space over F with ( v 1 , w 1 ) + ( v 2 , w 2 ) = ( v 1 + v 2 , w 1 + w 2 ) and c ( v, w ) = ( cv, cw ) Proof. We need to verify that Z is closed under addition and scalar multiplication and that Z with these operations satisfies the eight axioms of a vector space. Commutativity: ( v 1 , w 1 ) + ( v 2 , w 2 ) = ( v 1 + v 2 , w 1 + w 2 ) definition of addition in Z = ( v 2 + v 1 , w 2 + w 1 ) commutativity in V and W = (...
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This note was uploaded on 03/15/2012 for the course MATH 115A 262398211 taught by Professor Fuckhead during the Winter '10 term at UCLA.
 Winter '10
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