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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 xN-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?

*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 xN-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?

Advanced Engineering Mathematics
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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*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 xN-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 xN-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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