204Lecture92006 - Economics 204 Lecture 9Thursday, August...

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Economics 204 Lecture 9–Thursday, August 10, 2006 Revised 8/10/06, Revisions indicated by ** Section 3.3, Isomorphisms Defnition 1 Two vector spaces X, Y over a feld F are iso- morphic iF there is an invertible (recall this means one-to-one and onto) T L ( ). T is called an isomorphism . Isomorphic vector spaces are essentially indistinguishable as vector spaces. Theorem 2 (3.3) Two vector spaces over the same feld are isomorphic iF and only iF dim X =d im Y . ProoF: Suppose are isomorphic, via the isomorphism T . Let U = { u λ : λ Λ } be a basis oF X ,andlet v λ = T ( u λ ) ,V = { v λ : λ Λ } Since T is one-to-one, U and V are numerically equivalent. IF y Y , then there exists x X such that y = T ( x ) = T n X i =1 α λ i u λ i = n X i =1 α λ i T ( u λ i ) = n X i =1 α λ i v λ i 1
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which shows that V spans Y .T os e eth a t V is linearly indepen- dent, note that if 0= m X i =1 β i v λ i = m X i =1 β i T ( u λ i ) = T m X i =1 β i u λ i Since T is one-to-one, ker T = { 0 } ,so m X i =1 β i u λ i =0 Since U is a basis, we have β 1 = ··· = β m ,s o V is lin- early independent. Thus, V is a basis of Y ;s ince U and V are numerically equivalent, dim X =d im Y . Now suppose dim X Y .Le t U = { u λ : λ Λ } and V = { v λ : λ Λ } be bases of X and Y ; note we can use the same index set Λ for both because dim X Y . By Theorem 3.2, there is a unique T L ( X, Y ) such that T ( u λ )= v λ for all λ Λ. If T ( x )=0 , then T ( x ) = T n X i =1 α i u λ i = n X ı=1 α i T ( u λ i ) = n X ı=1 α i v λ i α 1 = = α n =0s V is a basis x 2
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ker T = { 0 } T is one-to-one If y Y , write y = m i =1 β i v λ i Let x = m X i =1 β i u λ i Then T ( x )= T m X i =1 β i u λ i = m X i =1 β i T ( u λ i ) = m X i =1 β i v λ i = y so T is onto, so T is an isomorphism and X, Y are isomorphic. Section 3.3 Supplement, Quotient Vector Spaces (not in de la Fuente): Defnition 3 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 ; this is a little weird at Frst, but ... . DeFne [ x ]+[ y ]=[ x + y ] α [ x αx ] 3
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You should check on your own that is an equivalence relation and that vector addition and scalar multiplication are well-defned, i.e.
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This note was uploaded on 03/06/2009 for the course ECON 0204 taught by Professor Staff during the Summer '08 term at University of California, Berkeley.

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204Lecture92006 - Economics 204 Lecture 9Thursday, August...

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