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Unformatted text preview: x )f ( y ) in two variables is always divisible by ( xy ), so g ( x,y ) is a polynomial. (d) Prove that the product rule D ( f ( x ) g ( x )) = f ( x ) D ( g ( x )) + D ( f ( x )) g ( x ) holds over any eld F . 2. Section 2.2, Exercise 2, parts (e,f,g). 3. Let T : V V be a linear transformation from a vector space V to itself. A subspace W V is called invariant for T if T ( W ) W . In this case, the restriction of T to the domain W is a linear transformation from W to itself, denoted T  W : W W . Let = { v 1 ,...,v n } be an ordered basis of V , and let W = Span( { v 1 ,...,v k } ). Prove that W is invariant for T if and only if the matrix [ T ] has the block form A B C , where A is a k k matrix, B is a k ( nk ) matrix, C is an ( nk ) ( nk ) matrix, and 0 denotes the ( nk ) k zero matrix. Also show that [ T  W ] { v 1 ,...,v k } = A ....
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 Fall '08
 GUREVITCH
 Linear Algebra, Algebra

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