Consider the vector space P2 of polynomials of degree at most 2 with real coefficients. Let S={-8x^2 + 4x – 5, -2x + 5). a. Give an example of a nonzero polynomial p(x) that is an element of span(S). p(x) = b. Give an example of a polynomial q(x) that is not an element of span(S). 9(x) = Note: if you receive the message "This answer is equivalent to the one you just submitted", please ignore it. The message is caused by a bug and has no meaning.

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22-08-2019
Mathematics

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Consider the vector space P2 of polynomials of degree at most 2 with real coefficients. Let S={-8x^2 + 4x – 5, -2x + 5). a. Give an example of a nonzero polynomial p(x) that is an element of span(S). p(x) = b. Give an example of a polynomial q(x) that is not an element of span(S). 9(x) = Note: if you receive the message "This answer is equivalent to the one you just submitted", please ignore it. The message is caused by a bug and has no meaning.

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LammettHash LammettHash · Answer 1 · 2019-08-22T00:59:57+02:00

a. Any vector in the span of [tex]S[/tex] is a linear combination of the vectors in [tex]S[/tex]. The simplest one we could come up with is the addition of the two vectors we know:

[tex]p(x)=(-8x^2+4x-5)+(-2x+5)=\boxed{-8x^2+2x}[/tex]

b. Since one vector is quadratic while the other is purely linear, there is no choice of [tex]c_1,c_2[/tex] such that

[tex]c_1(-8x^2+4x-5)+c_2(-2x+5)=\boxed{x}[/tex]

because the only way to eliminate the [tex]x^2[/tex] term is to pick [tex]c_1=0[/tex], but there's no way to eliminate the remaining constant term.