204PS4Solution2009 - Economics 204 Problem Set 4 Solutions Exercise 1 a First note that S is a subset of R3 which is a vector space(over R with the

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Economics 204 Problem Set 4 Solutions Exercise 1 a) First note that S is a subset of R 3 ; which is a vector space (over R ; with the operations assumed). Hence, all we have to show is that 0 vector is contained in S and that 8 ± 2 R , and x;y 2 S , we have + ±y 2 S . But this is pretty obvious: i ) take c = 0 to show that 0 2 S ; ii ) if x = c 1 v and y = c 2 v , then + ±y = ( 1 + ±c 2 ) v and if we let c = 1 + ±c 2 then it follows that + ±y 2 S: The space is one dimensional, and f v g is a basis for S . b) Same argument applies here: i ) 0 vector is obviously in S . ii ) Now take ± 2 R and x;y 2 S ; let z := + ±y , then z 1 + z 2 + z 3 = ( 1 + ±y 1 ) + ( 2 + ±y 2 ) + ( 3 + ±y 3 ) = ( x 1 + x 2 + x 3 ) + ± ( y 1 + y 2 + y 3 ) = 0 + ± 0 = 0; and z 1 + 2 z 2 = ( 1 + ±y 1 ) + 2( 2 + ±y 2 ) = ( 1 + 2 2 ) + ( 1 + 2 ±y 2 ) = ( x 1 + 2 x 2 ) + ± ( y 1 + 2 y 2 ) = 0 . The space is again one dimensional (Note that x 2 , then x 1 and x 3 are determined). f (1 ; 1 ; 0) g is a basis for S . c)
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This note was uploaded on 10/04/2010 for the course ECON 204 taught by Professor Anderson during the Summer '08 term at University of California, Berkeley.

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204PS4Solution2009 - Economics 204 Problem Set 4 Solutions Exercise 1 a First note that S is a subset of R3 which is a vector space(over R with the

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