College Algebra Exam Review 66

College Algebra Exam Review 66 - 76 1. ALGEBRAIC THEMES (c)...

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Unformatted text preview: 76 1. ALGEBRAIC THEMES (c) The set of real–valued functions on a set X , with pointwise addi- tion and multiplication of functions, .f C g/.x/ D f.x/ C g.x/ , and .fg/.x/ D f.x/g.x/ , for functions f and g and x 2 X , is a commutative ring. (d) The set of n-by- n matrices with integer entries is a noncommu- tative ring. There are many, many variations on these examples: matrices with complex entries or with with polynomial entries; functions with complex values, or integer values; continuous or differentiable functions; polyno- mials with any number of variables. Many rings have an identity element for multiplication , usually de- noted by 1 . (The identity element satisfies 1a D a1 D a for all elements a of the ring. Some rings do not have an identity element!) Example 1.11.3. (a) The identity matrix (diagonal matrix with diagonal entries equal to 1) is the multiplicative identity in the n-by- n matrices....
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This note was uploaded on 12/15/2011 for the course MAC 1105 taught by Professor Everage during the Fall '08 term at FSU.

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