- #1

saraaaahhhhhh

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Please explain how to solve this, if you help me. If you know of resources on how to think about this stuff, I'd greatly appreciate the assistance.

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Consider the collection of all polynomials (with complex coefficients) of degree less than N in x.

a.) Does this set constitutte 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^(N-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?

My attempt at a solution is:

a.) Yes, it doesw consitute a vector space. Any vector would be an ordered N-tuple (?) constructed from teh coefficients. How would I answer about the dimension of the space? Does it have N dimensions? I'm not sure if I understand what is being asked.

b.) Nothing changes?

c.) Then you'd have a pretty boring vector space? But I think all the rules would work.

d.) Still a vector space?

e.) Still a vector space? I don't see why that would change, I must be missing something.