ass3 - ASSIGNMENT 3 - MATH235, FALL 2007 Submit by 16:00,...

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ASSIGNMENT 3 - MATH235, FALL 2007 Submit by 16:00, Monday, October 1 (use the designated mailbox in Burnside Hall, 10 th floor). 1. The ring of 2 × 2 matrices over a field F . Let F be a field. (In a first read you may as well assume F is R , but solve the question in general). We consider the set M 2 ( F ) = ‰± a b c d : a,b,c,d F ² . It is called the two-by-two matrices over F . We define the addition of two matrices as ± a 1 b 1 c 1 d 1 + ± a 2 b 2 c 2 d 2 = ± a 1 + a 2 b 1 + b 2 c 1 + c 2 d 1 + d 2 . We define multiplication by ± a 1 b 1 c 1 d 1 ¶± a 2 b 2 c 2 d 2 = ± a 1 a 2 + b 1 c 2 a 1 b 2 + b 1 d 2 c 1 a 2 + d 1 c 2 c 1 b 2 + d 1 d 2 . Prove that this is a ring. For each of the following subsets of M 2 ( F ) determine if they are subrings or not. h A i The set ‰± a b 0 d M 2 ( F ) ² . h B i The set ‰± 0 b 0 0 M 2 ( F ) ² . h C i The set ‰± a 0 c d M 2 ( F ) ² . h D i The set ‰± a 0 0 d M 2 ( F ) ² . h E i The set ‰± a 0 0 a M 2 ( F ) ² . Remark: One defines in a very similar way the ring of n × n matrices with entries in a field F . 2. Find the quotient and remainder when a is divided by b : (1) a = 302 ,b = 19. (2) a = - 302 ,b = 19. (3) a
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This note was uploaded on 04/18/2008 for the course MATH 235 taught by Professor Goren during the Fall '07 term at McGill.

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