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.