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.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Question
c,d,b
Transcribed Image Text:1. Let n N and let V be the set of polynomials in the indeterminate with coefficients in R
of degree at most n. That is,
V = {anx" + an-1-¹+...+ 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
A (anx ++ a₁x+ao) = (Xan)x" +...+(Aa₁)x+ (Aao) 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) = anx² + ... + a₁x + ao,
B(p(x), q(x)) = [a¡bi
where
i=0
q(x) = bnxn ++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.
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