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Unformatted text preview: Supplement to Week 8 March 13, 2007 0.1 Algebras and Polynomials Definition: Let F be a field. An F-algebra is a vector space A endowed with a binary operation : A A A satisfying the following properties: 1. ( a b ) c = a ( b c ) for a, b, c A . 2. ( ra ) b = r ( a b ) = a ( r b ) for r F , a, b, A . 3. a ( b + c ) = a b + a c and ( b + c ) a = b a + c a for a, b, c A . 4. There is an element 1 A A such that 1 A a = a 1 A for all a A . An F-algebra is said to be commutative if in addition a b = b a for all a, b A . Examples: The field F , regarded as a vector space over itself, is a commutative F-algebra. If F = R is the field of real numbers, the field C of complex numbers is a commutative R-algebra. The set of n n matrices, with the usual operations, forms an F-algebra. It is not commutative if n > 1....
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