rev1_2011_04_05_03

rev1_2011_04_05_03 - EE263 S. Lall 2011.04.05.03 Problem...

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Unformatted text preview: EE263 S. Lall 2011.04.05.03 Problem Session 1 Solutions 1. Matrix representation of polynomial differentiation. 42035 We can represent a polynomial of degree less than n , p ( x ) = a n- 1 x n- 1 + a n- 2 x n- 2 + ··· + a 1 x + a , as the vector [ a a 1 ··· a n- 1 ] T ∈ R n . Consider the linear transformation D that differ- entiates polynomials, i.e. , D p = dp/dx . Find the matrix D that represents D ( i.e. , if the coefficients of p are given by a , then the coefficients of dp/dx are given by Da ). Solution. According to problem ?? it suffices to compute the transformation of the unit vectors e i ∈ R n for i = 1 , 2 , . . . , n under differentiation. In other words D = bracketleftbig De 1 De 2 ··· De n bracketrightbig . e 1 corresponds to the polynomial p 1 ( x ) = 1, and since D p 1 = 0 we have De 1 = . . . . e 2 corresponds to p 2 ( x ) = x with D p 2 = 1 and therefore De 2 = 1 . . . = e 1 . Similarly for p 3 ( x ) = x 2 we get D p 3 = 2 x so De 3 = 2 ....
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This note was uploaded on 08/06/2011 for the course EE 263 taught by Professor Boyd,s during the Summer '08 term at Stanford.

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rev1_2011_04_05_03 - EE263 S. Lall 2011.04.05.03 Problem...

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