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1. Let n e N and let V be the set of polynomials in the indeterminate x with coefficients in R of degree at most n. That is, V = {a,x" + an-1ª"-1+..+ a1x + ao : ao,a1,..., ɑn E R}. (a) Show that V is a vector space over R under addition of polynomials and scalar multipli- cation defined by A (ana" + ... + a1x + ao) = (Aan)æ" + ..+ (Aa1)x + (Aao) for AeR. (b) Calculate the dimension of V by finding a basis and justify your answer. (c) We define a scalar product 3 on V by p(x) = a„x" + · ·.+a1x+ ao; B(p(x), q(x)) = E a;bi where i=0 q(x) bnx" + · ..+ bịx + bo. Prove that 3 is a positive definite scalar product.

1. Let n e N and let V be the set of polynomials in the indeterminate x with coefficients in R of degree at most n. That is, V = {a,x" + an-1ª"-1+..+ a1x + ao : ao,a1,..., ɑn E R}. (a) Show that V is a vector space over R under addition of polynomials and scalar multipli- cation defined by A (ana" + ... + a1x + ao) = (Aan)æ" + ..+ (Aa1)x + (Aao) for AeR. (b) Calculate the dimension of V by finding a basis and justify your answer. (c) We define a scalar product 3 on V by p(x) = a„x" + · ·.+a1x+ ao; B(p(x), q(x)) = E a;bi where i=0 q(x) bnx" + · ..+ bịx + bo. Prove that 3 is a positive definite scalar product.

Linear Algebra: A Modern Introduction
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter6: Vector Spaces
Section6.1: Vector Spaces And Subspaces
Problem 19EQ
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1. Let n eN and let V be the set of polynomials in the indeterminate r with coefficients in R of degree at most n. That is, V = {anx" + an-12"-1+ ..+ a1x + ao : ao, a1,...,an E R}. (a) Show that V is a vector space over R under addition of polynomials and scalar multipli- cation defined by A (ana" + ...+ a1x + ao) = (Aan)" + ...+ (Aa1)x + (Aao) for AeR. (b) Calculate the dimension of V by finding a basis and justify your answer. (c) We define a scalar product 3 on V by p(x) anx" + ...+ a1x + ao, n B(p(x), q(x)) = Da;bi where q(x) bnr" + ...+ b1x+ bo. Prove that 3 is a positive definite scalar product. (d) Let n = 2 and use the Gram-Schmidt Process (or otherwise) to find an orthogonal basis for V containing the vector v1 =1+x+x².
Transcribed Image Text:1. Let n eN and let V be the set of polynomials in the indeterminate r with coefficients in R of degree at most n. That is, V = {anx" + an-12"-1+ ..+ a1x + ao : ao, a1,...,an E R}. (a) Show that V is a vector space over R under addition of polynomials and scalar multipli- cation defined by A (ana" + ...+ a1x + ao) = (Aan)" + ...+ (Aa1)x + (Aao) for AeR. (b) Calculate the dimension of V by finding a basis and justify your answer. (c) We define a scalar product 3 on V by p(x) anx" + ...+ a1x + ao, n B(p(x), q(x)) = Da;bi where q(x) bnr" + ...+ b1x+ bo. Prove that 3 is a positive definite scalar product. (d) Let n = 2 and use the Gram-Schmidt Process (or otherwise) to find an orthogonal basis for V containing the vector v1 =1+x+x².
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