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

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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 ² g ) and thus dim Im g ³ dim Im( f ² 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 0 g . c) Let Z = W = V . Show that if f; g are autmorphisms of V (i.e. isomor- phisms from V to V ), then f ² g is an automorphism of V . Exercise 3 a) What 2 by 2 matrix represents, with respect to the standard basis, the transformation which rotates every vector in R 2 counterclockwise 90
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