Lecture Notes - Mth 543

Lecture Notes - Mth 543 - MTH 543 - ABSTRACT LINEAR ALGEBRA...

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Unformatted text preview: MTH 543 - ABSTRACT LINEAR ALGEBRA OREGON STATE UNIVERSITY FALL 2011 TEXT: NONE L ECTURE N OTES Vector Spaces Definition: Let F be a set with at least two elements, and with two binary operations, + , . Then F is a field under these operations if (1) F is an abelian group under + , with additive identity . (2) F \ { } is an abelian group under , with multiplicative identity 1 . (3) Distributive law holds: 1 ( 2 + 3 ) = 1 2 + 1 3 , 1 , 2 , 3 F . Recall: G is a group under * if (1) G is closed under * . That is, * G whenever , G . (2) There is an identity, e , such that * e = e * = , G . (3) Associative: * ( * ) = ( * ) * , ,, G . (4) There is an inverse, - 1 , for each G . That is, * - 1 = - 1 * = e, G . And G is abelian if commutativity holds. Examples : 1. R is a field under the usual + , . 2. C is a field under the usual + , . 3. Q is a field under the usual + , . Note that Q R C is a tower of fields (subset plus same operations under restrictions). 4. F p = Z p , the field of integers mod p , where p is prime. Definition: Let F be a field under + and . Then the characteristic of F is the least positive integer n such that 1 + 1 + + 1 | {z } n-times = 0 . If no such n exists, then the characteristic is . Examples : 1. char ( C ) = char ( R ) = char ( Q ) = 0 . 1 2. char ( F p ) = p , if p is prime. 3. if char ( F ) = n = a b is not prime, then 1 < a,b < n and 0 = 1 + 1 + + 1 | {z } n-times = (1 + 1 + + 1 | {z } a-times )(1 + 1 + + 1 | {z } b-times ) . Therefore, we have the product of two elements of F equal to , which means that one of the elements is . Without loss of generality, 1 + 1 + + 1 | {z } a-times = 0 . But a < n , a contradiciton to n being the characteristic of F . Definition: Let V 6 = , let F be a field, and let + : V V V , and : F V V be functions. Then V is a vector space over F if (1) V is an abelian group under + . (2) 1 v = v, v V and ( ) v = ( v ) (3) Distribituve laws hold: ( v + w ) = v + w, F , v,w V, ( + ) v = v + v, , F , v V. Examples: Vector spaces over a field F . 1. F n = { ( 1 , 2 ,..., n ) : i F , i = 1 , 2 ,...,n } 2. M m n ( F ) , m n matrices with entries in F . 3. Let n N . Then P n ( F ) , the set of polynomials with degree at most n and coeffi- cients in F , and the polynomial, is a vector space over F . 4. The set of all polynomials, P ( F ) . 5. Let V be a vector space over F , and let S be any non-empty set. Let F ( S,V ) = { f : S V, f is a function } , and define ( f 1 + f 2 )( s ) = f 1 ( s ) + f 2 ( s ) , f 1 ,f 2 F , and ( f )( s ) = ( f ( s )) , F , f F ....
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Lecture Notes - Mth 543 - MTH 543 - ABSTRACT LINEAR ALGEBRA...

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