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*Problem 3.2 Consider the collection of all polynomials (with complex coefficients) of degree N in x. (a) Does this set constitute a vector space (with the polynomials as "vectors")? If so, suggest a convenient basis, and give the dimension of the space. If not, which of the defining properties does it lack? (b) What if we require that the polynomials be even functions? (c) What if we require that the leading coefficient (i.e., the number multiplying

*Problem 3.2 Consider the collection of all polynomials (with complex coefficients) of degree N in x. (a) Does this set constitute a vector space (with the polynomials as "vectors")? If so, suggest a convenient basis, and give the dimension of the space. If not, which of the defining properties does it lack? (b) What if we require that the polynomials be even functions? (c) What if we require that the leading coefficient (i.e., the number multiplying

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter4: Polynomial And Rational Functions
Section: Chapter Questions
Problem 15RE
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*Problem 3.2 Consider the collection of all polynomials (with complex coefficients) of degree < N in x. (a) Does this set constitute a vector space (with the polynomials as "vectors")? If so, suggest a convenient basis, and give the dimension of the space. If not, which of the defining properties does it lack? (b) What if we require that the polynomials be even functions? (c) What if we require that the leading coefficient (i.e., the number multiplying x-1) be 1? (d) What if we require that the polynomials have the value 0 at x = 1? (e) What if we require that the polynomials have the value 1 at x = = 0?
Transcribed Image Text:*Problem 3.2 Consider the collection of all polynomials (with complex coefficients) of degree < N in x. (a) Does this set constitute a vector space (with the polynomials as "vectors")? If so, suggest a convenient basis, and give the dimension of the space. If not, which of the defining properties does it lack? (b) What if we require that the polynomials be even functions? (c) What if we require that the leading coefficient (i.e., the number multiplying x-1) be 1? (d) What if we require that the polynomials have the value 0 at x = 1? (e) What if we require that the polynomials have the value 1 at x = = 0?
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