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K[T denotes te ring of polynomials over a field K and K ((x) denotes the ring of formalpower series (that is, ieN Cwhere c E K, no convergence is required for the series, andmultiplication 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 asvector spaces over K(c) Let W be an arbitrary vector space over K. Show that, for every function 7: N Kthere is a unique linear map f: K [a] -> W so that f(x') = T(i) for all i E N.

Question

Please, help me with a very detailed and self-explanatory solution to this problem. I will appreciate it alot. Thank you

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.
help_outline

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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.

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check_circleAnswer
Step 1

Problem concerns the ring and vector space structure of the space of formal power series over K, K a field

Step 2

(a) Here K[x] is the ring of polynomials over K ; note that any polynomial consisists only finitely many terms . Thus , as a vector space ,K[x] is isomorphic to the space of sequences over K which are eventually 0, (as described in the last two lines

(a)K[x] has basis 1,x, x ,...x",
Every f(x)e K[x] has finite deg ree,
so it is a finite linear combination of
Hence.K[x]is isomorphic to K).
K () The set of all sequences
{(a,):m, with a,
0
help_outline

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(a)K[x] has basis 1,x, x ,...x", Every f(x)e K[x] has finite deg ree, so it is a finite linear combination of Hence.K[x]is isomorphic to K). K () The set of all sequences {(a,):m, with a, 0

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Step 3

(a) in contrast, K<<x>> is the ring of formal power series , here an element of K<<x>> can have infinitely many non-zero coefficients....

(a)Now, K << x» consists of all
linear combinationsax",with
no conditions on a 's
71
For example,
x"eK <<x>»
but not to K[x], as it does not have finite deg ree
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(a)Now, K << x» consists of all linear combinationsax",with no conditions on a 's 71 For example, x"eK <<x>» but not to K[x], as it does not have finite deg ree

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