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Chapter 1: Problem 8

Show that \(P_{n}(F)\) is generated by \(\left\\{1, x, \ldots, x^{n}\right\\}\).

Short Answer

Expert verified
The polynomial function \(P_{n}(F)\) of degree \(n\) over a field \(F\) can be written as a linear combination of the set \(\{1, x, \ldots, x^{n}\}\): \[P_{n}(F) = a_{0}(1) + a_{1}(x) + a_{2}(x^{2}) + \cdots + a_{n}(x^{n})\] where \(a_{0}, a_{1}, \ldots, a_{n} \in F\) are coefficients of the polynomial. Thus, we have shown that \(P_{n}(F)\) is generated by the set \(\{1, x, \ldots, x^{n}\}\).

Step by step solution

01

Define the polynomial function

A polynomial function of degree \(n\) over a field \(F\) can be written as: \[P_{n}(F) = a_{0} + a_{1}x + a_{2}x^{2} + \cdots + a_{n}x^{n}\] where \(a_{0}, a_{1}, \ldots, a_{n} \in F\) are coefficients of the polynomial.
02

Rewrite the polynomial function using the set

We can express the given polynomial function as a linear combination of the set \(\{1, x, \ldots, x^{n}\}\). This can be done as follows: \[P_{n}(F) = a_{0}(1) + a_{1}(x) + a_{2}(x^{2}) + \cdots + a_{n}(x^{n})\]
03

Prove that polynomials are generated by the set

From Step 2, we have expressed the polynomial function \(P_{n}(F)\) in terms of the set \(\{1, x, \ldots, x^{n}\}\). Since every polynomial can be written as a linear combination of these elements, the set generates the polynomials. Hence, we have shown that \(P_{n}(F)\) is generated by the set \(\{1, x, \ldots, x^{n}\}\).

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