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Unformatted text preview: 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 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....
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