Section33supplementTimeless - Econ 204 Supplement to...

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Econ 204 Supplement to Section 3.3 The Matrix Representation of a Linear Transformation The purpose of this note is to provide a brief treatment of quotient vector spaces, and amplify on the relationship between linear transformations and their matrix representations. DeFnition 1 Given a vector space X and a vector subspace W X , deFne an equivalence relation by x y x y W Exercise 2 Show that is an equivalence relation. DeFnition 3 We deFne a new vector space V/W ,the quotient of V W . The set of vectors in is { [ x ]: x X } where we recall that [ x ] is the equivalence class of x with respect to the equivalence relation . In other words, [ x ]= { y X : x y W } = { x + w : w W } Thus, each of the vectors is a set ; this is a little weird at Frst, but try to get used to it. Now, we have to deFne the operations of vector addition and scalar multiplication. The deFnitions are [ x ]+[ y ]=[ x + y ] α [ x αx ] One needs to check that these deFnitions make sense. [ x ] is a set, and there are potentially many di±erent representatives, i.e. many x 0 such that [ x [ x 0
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This note was uploaded on 08/01/2008 for the course ECON 204 taught by Professor Anderson during the Fall '08 term at Berkeley.

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Section33supplementTimeless - Econ 204 Supplement to...

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