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: α 1 ( α 2 + α 3 ) = α 1 α 2 + α 1 α 3 , ∀ α 1 ,α 2 ,α 3 ∈ F . Recall: G is a group under * if (1) G is closed under * . That is, α * β ∈ G whenever α,β ∈ G . (2) There is an identity, e , such that α * e = e * α = α, ∀ α ∈ G . (3) Associative: α * ( β * γ ) = ( α * β ) * γ, ∀ α,β,γ ∈ G . (4) There is an inverse, α- 1 , for each α ∈ G . That is, α * α- 1 = α- 1 * α = e, ∀ α ∈ 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 ( αβ ) · v = α · ( β · v ) (3) Distribituve laws hold: α · ( v + w ) = α · v + α · w, ∀ α ∈ F , ∀ v,w ∈ V, ( α + β ) · v = α · v + β · v, ∀ α,β ∈ F , ∀ v ∈ V. Examples: Vector spaces over a field F . 1. F n = { ( α 1 ,α 2 ,...,α n ) : α 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 ( α · f )( s ) = α · ( f ( s )) , ∀ α ∈ F , ∀ f ∈ F ....
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Lecture Notes - Mth 543 - MTH 543 ABSTRACT LINEAR ALGEBRA...

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