Question

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.

Answer 1

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

$p(x)=(-8x^2+4x-5)+(-2x+5)=\boxed{-8x^2+2x}$

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

$c_1(-8x^2+4x-5)+c_2(-2x+5)=\boxed{x}$

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