Week5 - Supplement to Week 5 0.1 Duality Here are the key...

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Supplement to Week 5 February 20, 2007 0.1 Duality Here are the key definitions and formulas. This material is also in the book. Definition: If W is a vector space over a field F and X is any set, F ( X, W ) is the set of functions from X to W . If f and g are elements of F ( X, W ), then f + g ∈ F ( X, W ) is by definition the function such that ( f + g )( x ) = f ( x ) + g ( x ) for all x X, and if c F , cf ∈ F ( X, W ) is by definition the function such that cf ( x ) = c ( f ( x )) for all x X. One checks easily that with these definitions, F ( X, W ) is a vector space over F . Proposition: With the definition and notation above, the following hold. 1. If x X , then the mapping x : F ( X, W ) W : x ( f ) := f ( x ) is a linear transformation. 2. Suppose that V and W are vector spaces over F . Then the space L ( V, W ) of linear transformations V W is a linear subspace of F ( V, W ). Be sure you can write proofs of these statements. 1
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Definition: If V is a vector space, then its dual is the vector space V * := L ( V, F ) . For example, if V = F n , then for each i , the mapping π i : V F sending ( x 1 , . . . , x n ) to x i (the i th coordinate of ( x 1 , . . . x n )) is linear, hence belongs to V * . If α := ( v 1 , . . . v n ) is an ordered basis for V , then for each v V there is a unique φ α ( v ) := ( a 1 , . . . a n ) F n such that v = i a i v i . The map φ α : V F n , is an isomorphism, and for each i , the map φ i : V F sending v to a i
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