Let n € 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 = {anx" + an-1x²-¹ +...+ a₁x + ao ao, a₁,..., an ER}. (a) Show that V is a vector space over R under addition of polynomials and scalar multipli- cation defined by X(anx ++ a1₁x + ao) = (λan)x² + ... + (№a₁)x+ (ao) for XER. (b) Calculate the dimension of V by finding a basis and justify your answer. (c) We define a scalar product 3 on V by n p(x) = anx" + + a₁x+ao, β(p(x),q(x)) = Σ aibi where i=0 q(x) = b₂x +...+b₁x+bo. Prove that 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.

Let n € 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 = {anx" + an-1x²-¹ +...+ a₁x + ao ao, a₁,..., an ER}. (a) Show that V is a vector space over R under addition of polynomials and scalar multipli- cation defined by X(anx ++ a1₁x + ao) = (λan)x² + ... + (№a₁)x+ (ao) for XER. (b) Calculate the dimension of V by finding a basis and justify your answer. (c) We define a scalar product 3 on V by n p(x) = anx" + + a₁x+ao, β(p(x),q(x)) = Σ aibi where i=0 q(x) = b₂x +...+b₁x+bo. Prove that 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.

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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