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 ﬁeld 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 β = { 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 ] β has the block form ± A B C ² , 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, Derivative, Vector Space, Haiman Problem Set, twoelement ﬁeld, unique linear operator

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