Math 220 Assignment

November 16, 2001

Due Monday, November 19

1. Let P_{3} be the vector space of polynomials of degree at most 3, and let phi be the linear map from P_{3} to itself that is defined by the formula

   ( phi(f) )

   (x)   =   INT[_{0}^{x} f'(t) d t ] ,

   where f' denotes the derivative of f. Find the matrix of phi with respect to the basis of P_{3} given by the powers of the variable.

2. Let P_{2} be the vector space of polynomials of degree at most 2. Define a scalar product (analogous to ``dot'' product) Gamma on P_{2} with the formula

   Gamma(f, g)  =  INT[_{0}^{1} f(t) g(t) dt ] .

   Find the orthogonal complement, relative to Gamma, of the subspace consisting of the constant polynomials.

3. Use the Gram-Schmidt process (§ 3.4 of the text) to make an orthonormal basis for P_{2}, relative to the scalar product Gamma of the previous exercise, containing the constant polynomial 1.

4. Repeat the previous exercise using the inner product

   Delta(f, g)  =  {1}/{2} INT[_{-1}^{1} f(t) g(t) dt ] .

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