# K[T denotes te ring of polynomials over a field K and K ((x) denotes the ring of formal power series (that is, ieN Cwhere c E K, no convergence is required for the series, and multiplication is performed just like polynomials) (a) Show that, as vector spaces, Kz] is isomorphic to K0N9) and K (x') is isomorphic to KN |(b) Explain why the set {r i E N} is a basis for Kr] but not a basis for K ((r), both as vector spaces over K (c) Let W be an arbitrary vector space over K. Show that, for every function 7: N K there is a unique linear map f: K [a] -> W so that f(x') = T(i) for all i E N.

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Please, help me with a very detailed and self-explanatory solution to this problem. I will appreciate it alot. Thank you help_outlineImage TranscriptioncloseK[T denotes te ring of polynomials over a field K and K ((x) denotes the ring of formal power series (that is, ieN Cwhere c E K, no convergence is required for the series, and multiplication is performed just like polynomials) (a) Show that, as vector spaces, Kz] is isomorphic to K0N9) and K (x') is isomorphic to KN |(b) Explain why the set {r i E N} is a basis for Kr] but not a basis for K ((r), both as vector spaces over K (c) Let W be an arbitrary vector space over K. Show that, for every function 7: N K there is a unique linear map f: K [a] -> W so that f(x') = T(i) for all i E N. fullscreen

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