204Lecture92009web

204Lecture92009web - Economics 204 Lecture 9Thursday,...

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Economics 204 Lecture 9–Thursday, August 6, 2009 Section 3.3 Supplement, Quotient Vector Spaces (not in de la Fuente): Defnition 1 Given a vector space X and a vector subspace W of X , deFne an equivalence relation by x y x y W ±orm a new vector space X/W : the set of vectors is { [ x ]: x X } where [ x ] denotes the equivalence class of x with respect to . Note that the vectors are sets ;th isisa little weird at Frst, but . . . . DeFne [ x ]+[ y ]=[ x + y ] α [ x αx ] You should check on your own that is an equivalence relation and that vector addition and scalar multiplication are well-deFned, i.e. [ x x 0 ] , [ y y 0 ] [ x + y x 0 + y 0 ] [ x x 0 ] F [ αx αx 0 ] Theorem 2 If dim X< ,then dim ( X/W )=d im X dim W Theorem 3 Let T L ( X, Y ) .Then Im T is isomorphic to X/ ker T . 1
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Proof: If dim X< ,thend im X/ ker T =d X dim ker T (from the previous theorem) = Rank T (from Theorem 11 in yesterday’s lecture) = dim Im T ,so X/ ker T is isomorphic to Im T . We shall prove that it is true in general, and that the isomorphism is natural. DeFne ˜ T ([ x ]) = T ( x ) We need to check that this is well-deFned. [ x ]=[ x 0 ] x x 0 x x 0 ker T T ( x x 0 )=0 T ( x )= T ( x 0 ) so ˜ T is well-deFned. Clearly, ˜ T : X/ ker T Im T . It is easy to check that ˜ T is linear, so ˜ T L ( X/ ker T, Im T ). ˜ T ([ x ]) = ˜ T ([ y ]) T ( x T ( y ) T ( x y x y ker T x y [ x y ] so ˜ T is one-to-one. y Im T ⇒∃ x X T ( x y ˜ T ([ x ]) = y so ˜ T is onto, hence ˜ T is an isomorphism. Example: Consider T L ( R 3 , R 2 ) deFned by T ( x, y, z )=( x, y ) 2
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Then ker T = { ( x, y, z ) R 3 : x = y =0 } is the z -axis. Given ( x, y, z ), the equivalence class [( x, y, z )] is just the line through ( x, y, 0) parallel to the z -axis. ˜ T ([( x, y, z )]) = T ( x, y, z )=( x, y ). Back to de la Fuente: Every real vector space X with dimension n is isomorphic to R n . What’s the isomorphism? Defnition 4 Fix any Hamel basis V = { v 1 ,...,v n } of X .Any x X has a unique representation x = n X j =1 β j v j (here, we allow β j = 0). Generally, vectors are represented as column vectors, not row vectors. crd V ( x )= β 1 . . . β n R n crd V ( x ) is the vector of coordinates of x with respect to the basis V . crd V ( v 1 1 0 .
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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 Berkeley.

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204Lecture92009web - Economics 204 Lecture 9Thursday,...

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