week8 - Supplement to Week 8 0.1 Algebras and Polynomials...

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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. If V is a vector space, the set End ( V ) of linear maps V V , with the vector space structure of L ( V, V ) and composition as the binary operation · , is an F -algebra.
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