204PS42009

204PS42009 - Economics 204 Problem Set 4 Due Tuesday,...

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Unformatted text preview: Economics 204 Problem Set 4 Due Tuesday, August 11 Exercise 1 State (and check) whether each of the following is a vector space (over R ). a) S = f cv : c 2 R; v = (1 ; 1 ; 1) g b) S = f ( x 1 ;x 2 ;x 3 ) : x 1 + x 2 + x 3 = 0 ; x 1 + 2 x 2 = 0 g c) S = f ( x 1 ;x 2 ) : x 1 + x 2 = 1 g d) S = f f : [0 ; 1] ! [0 ; 1] : f continuous g . (&rst, de&ne ( f + g )( x ) := f ( x ) + g ( x ) and ( cf )( x ) = cf ( x ) ). If to any one of ( a ) & ( c ) you answered "yes", then &nd the dimension of the space and a Hamel basis for it. Exercise 2 Let Z;V;W be vector spaces and g : Z ! V , f : V ! W be linear transfor- mations and Z;V; and W have dimension n . a) Show that Ker ( g ) Ker ( f g ) and thus dim Im g dim Im( f g ) , where dim Im g indicates the dimension of the image of the map g , and Ker ( h ) = f x 2 V : h ( x ) = 0 g . b) Show that f is one-to-one if and only if Ker ( f ) = f g ....
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204PS42009 - Economics 204 Problem Set 4 Due Tuesday,...

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