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.

Question

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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.

Image Transcription

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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