hw04 - x-f y in two variables is always divisible by x-y so...

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Math 110—Linear Algebra Fall 2009, Haiman Problem Set 4 Due Monday, Sept. 28 at the beginning of lecture. 1. If F is any field, and n is a non-negative integer, we can define a corresponding element of F to be the sum 1 + 1 + ··· + 1 in F with n terms. (We usually denote this element by n as well, although strictly speaking this is bad notation, since the integer n and the element n of F are two different things.) For example, in the two-element field F 2 , 0 stands for 0, 1 for 1, 2 for 1 + 1 = 0, 3 for 2 + 1 = 1, and so on: for n even we will have n = 0, and for n odd, n = 1 . Now let D : P ( F ) P ( F ) be the unique linear operator whose values on the basis of monomials are given by D (1) = 0 and D ( x n ) = nx n - 1 for n > 0. (a) Show that for F = R , D is differentiation, that is, D ( f ( x )) = f 0 ( x ) for all f ( x ) P ( R ). (b) If F = F 2 is the two-element field, find a simple formula for the “second derivative” D ( D ( f ( x )). (c) Letting F be arbitrary once again, prove that for all f ( x ) P ( F ), we have D ( f ( x )) = g ( x,x ), where g ( x,y ) = ( f ( x ) - f ( y )) / ( x - y ). You may assume without proof the fact that the polynomial f (
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Unformatted text preview: x )-f ( y ) in two variables is always divisible by ( x-y ), 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 × ( n-k ) matrix, C is an ( n-k ) × ( n-k ) matrix, and 0 denotes the ( n-k ) × k zero matrix. Also show that [ T | W ] { v 1 ,...,v k } = A ....
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