Prove that the set Pn of all polynomials of degree less than or equal to n forms a vector space under the "usual" addition and scalar multiplication.
Prove that the set Pn of all polynomials of degree less than or equal to n forms a vector space under the "usual" addition and scalar multiplication.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![Prove that the set \( P_n \) of all polynomials of degree less than or equal to \( n \) forms a vector space under the "usual" addition and scalar multiplication.
- Note that this means that every axiom in the definition of a vector space needs to be justified. For example, here is justification of the closure under scalar multiplication property:
Let \( p(x) \in P_n \) and \( \alpha \) a scalar. Then \( p(x) = a_n x^n + a_{n-1} x^{n-1} + \cdots + a_1 x + a_0 \) for some scalars \( a_0, \ldots, a_n \). Then
\[
\begin{align*}
\alpha p(x) &= \alpha (a_n x^n + a_{n-1} x^{n-1} + \cdots + a_1 x + a_0) \\
&= (\alpha a_n) x^n + (\alpha a_{n-1}) x^{n-1} + \cdots + (\alpha a_1) x + \alpha a_0
\end{align*}
\]
is a polynomial of degree at most \( n \) so that \( \alpha p(x) \in P_n \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fb04829d0-4645-426e-bf1a-7ada40b0786f%2F5fd1324f-6306-4547-b8f7-e099ab1d6b8e%2F9jfw6up_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Prove that the set \( P_n \) of all polynomials of degree less than or equal to \( n \) forms a vector space under the "usual" addition and scalar multiplication.
- Note that this means that every axiom in the definition of a vector space needs to be justified. For example, here is justification of the closure under scalar multiplication property:
Let \( p(x) \in P_n \) and \( \alpha \) a scalar. Then \( p(x) = a_n x^n + a_{n-1} x^{n-1} + \cdots + a_1 x + a_0 \) for some scalars \( a_0, \ldots, a_n \). Then
\[
\begin{align*}
\alpha p(x) &= \alpha (a_n x^n + a_{n-1} x^{n-1} + \cdots + a_1 x + a_0) \\
&= (\alpha a_n) x^n + (\alpha a_{n-1}) x^{n-1} + \cdots + (\alpha a_1) x + \alpha a_0
\end{align*}
\]
is a polynomial of degree at most \( n \) so that \( \alpha p(x) \in P_n \).
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