235FA10-SAMPLEM1-sol - MTH 235 Linear Algebra Sample problems Midterm 1 1(a Argue that Z/5Z is a eld by using addition and multiplication tables(b

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MTH 235 : Linear Algebra Sample problems - Midterm 1 1. (a) Argue that Z / 5 Z is a field by using addition and multiplication tables. (b) Argue that Z / 4 Z is not a field. Solution. (a) The addition and multiplication is inherited from that of the inte- gers Z , so it satisfies most of the axioms of a field. We need to show existence of additive/multiplicative identity and additive/multiplicative inverses. + 0 1 2 3 4 0 0 1 2 3 4 1 1 2 3 4 0 2 2 3 4 0 1 3 3 4 0 1 2 4 4 0 1 2 3 · 0 1 2 3 4 0 0 0 0 0 0 1 0 1 2 3 4 2 0 2 4 1 3 3 0 3 1 4 2 4 0 4 3 2 1 The additive identity is 0, and - 1 = 4, - 2 = 3, - 3 = 2, - 4 = 1 so every element has an additive inverse. The multiplicative identity is 1 and 2 - 1 = 3, 3 - 1 = 2, 4 - 1 = 4 so that every element has a multiplicative inverse. (b) The tables for Z / 4 Z are: + 0 1 2 3 0 0 1 2 3 1 1 2 3 0 2 2 3 0 1 3 3 0 1 2 · 0 1 2 3 0 0 0 0 0 1 0 1 2 3 2 0 2 0 2 3 0 3 2 1 While Z / 4 Z has all its additive inverses ( - 1 = 3, - 2 = 2, - 3 = 1) it does not have all of its multiplicative inverses since there is no element a Z / 4 Z so that 2 · a = 1. 1
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2. Let V be the vector space of functions f : R R . Show that W V is a subspace, where (a) W = { f | f (1) = 0 } (b) W = { f | f (3) = f (1) } (c) W = { f | f ( - x ) = - f ( x ) } Solution. Need only to show that W is non-empty and if α,β W and c R then + β W . (a) The constant function f ( x ) = 0 is in W . Now suppose that f,g W , i.e., that f (1) = 0 = g (1). We want to show that cf + g W , i.e., that ( cf + g )(1) = 0. ( cf + g )(1) = cf (1) + g (1) = c · 0 + 0 = 0 so W is a subspace. (b) Once again, any constant function is in W . Now assume that f,g W , i.e., that f (1) = f (3) and g (1) = g (3): ( cf + g )(1) = cf (1) + g (1) = cf (3) + g (3) = ( cf + g )(3) so W is a subspace. (c) The function f ( x ) = sin( x ) is in this set. Suppose that f,g W , i.e., that f ( - x ) = - f ( x ) and g ( - x ) = - g ( x ): ( cf + g )( - x ) = cf ( - x ) + g ( - x ) = - cf ( x ) - g ( x ) = - ( cf + g )( x ) so W is a subspace. 2
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3. Prove that W 0 = { ( a 1 ,a 2 ,...,a n ) F n | a 1 + a 2 + ··· + a n = 0 } is a subspace of F n , but W 1 = { ( a 1 ,a 2 ,...,a n ) F n | a 1 + a 2 + ··· + a n = 1 } is not. Solution.
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This note was uploaded on 11/14/2010 for the course PHYCS 498 taught by Professor Aa during the Spring '10 term at University of Illinois, Urbana Champaign.

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235FA10-SAMPLEM1-sol - MTH 235 Linear Algebra Sample problems Midterm 1 1(a Argue that Z/5Z is a eld by using addition and multiplication tables(b

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