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.
Correlation
Correlation defines a relationship between two independent variables. It tells the degree to which variables move in relation to each other. When two sets of data are related to each other, there is a correlation between them.
Linear Correlation
A correlation is used to determine the relationships between numerical and categorical variables. In other words, it is an indicator of how things are connected to one another. The correlation analysis is the study of how variables are related.
Regression Analysis
Regression analysis is a statistical method in which it estimates the relationship between a dependent variable and one or more independent variable. In simple terms dependent variable is called as outcome variable and independent variable is called as predictors. Regression analysis is one of the methods to find the trends in data. The independent variable used in Regression analysis is named Predictor variable. It offers data of an associated dependent variable regarding a particular outcome.
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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.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F0282592e-adf8-40c1-9d31-7bca34586c75%2F0eea8960-6870-4161-99dc-a0871bdd56dd%2F7yduc2d.png&w=3840&q=75)
Problem concerns the ring and vector space structure of the space of formal power series over K, K a field
(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](https://content.bartleby.com/qna-images/answer/0282592e-adf8-40c1-9d31-7bca34586c75/1e5ccb55-7d32-4363-a6ed-7a80b501a239/e4cjw67.png)
(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. So, as a vector space K<<x>> is isomorphic to the space of ALL sequences over K, with no restrictions (unlike in the case of the ring K[x]
![(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](https://content.bartleby.com/qna-images/answer/0282592e-adf8-40c1-9d31-7bca34586c75/1e5ccb55-7d32-4363-a6ed-7a80b501a239/ymscfil.png)
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