Department of Mathematics, K.T.H.M. College, Nashik

mathsbeauty

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Registered: 11-06-2015

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Vector Space of Polynomials (Part C Question: NET June-2015)

(Part C Question: NET June-2015) Let \(p_n(x) = x^n \) for \(x \in \mathbb{R} \) and let  \(\mathcal{P} = \) span \(\{p_0,p_1,\ldots\} \). Then
A. \(\mathcal{P} \) is the vector space of all real valued continuous functions on \(\mathbb{R} \).
B. \(\mathcal{P} \) is a subspace of all real valued continuous functions on \(\mathbb{R} \).
C. \(\{p_0,p_1, \ldots \} \) is a linearly independent set in the vector space of all continuous functions on \(\mathbb{R} \)
D. Trigonometric functions belongs to \(\mathcal{P} \).

\(\textbf{Options B and C}\)
Option A: Not every continuous real valued function on \(\mathbb{R} \) is a polynomial.
Option B: All polynomials in \(\mathbb{R}[x] \) are continuous and they are closed under addition and scalar multiplication.
Option C: \(a_0+a_1x+\cdots+a_nx^n = 0 \iff a_0=a_1=\ldots=a_n=0 \)
Option D: Trigonometric functions are not polynomials!