Subspace Test with P3 - A vector space of polynomials of degree at most 3

by Chad   Last Updated July 23, 2019 20:20 PM

Show that W = {p ∈ P3 : p(2) = 0}, the set of polynomials in P3 with p(2) = 0, is a subspace of P3.

Answers 1

Are you taking it as given that P2 is a linear space? If so then you only need to prove that the subset of polynomials such that p(2)= 0 is "closed" under addition and multiplication by scalars (numbers).

Suppose p1 and p2 are polynomials, of degree 3 or less, such that p1(2)= 0 and p2(2)= 0. What is true of the polynomial p1+ p2? What is true of the polynomial ap1 where a is a scalar?

(Do you see why, if we changed the condition from "p(2)= 0" to "p(0)= 2", this would not be a subpace?)

July 23, 2019 20:16 PM

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