Unformatted text preview: 2.4.15 ⇒ : Suppose that T ( β ) is a basis for W . Then R ( T ) = span( T ( β )) = W , so T is onto. Since dim( V ) = dim( W ), T is also onetoone and thus an isomorphism. ⇐ : Suppose that T is an isomorphism. Then, since T is onetoone, we have proved (previous hw) that T ( β ) is linearly independent. Since T ( β ) has dim( W ) elements, it must be a basis for W ....
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This note was uploaded on 11/30/2009 for the course MATH 115A taught by Professor Liu during the Winter '07 term at UCLA.
 Winter '07
 Liu
 Linear Algebra, Algebra, Addition

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