How to check if a polynomial equation belongs to a span

[alexngo]

hey
i would greatly appreciate the solutions for the question below

1) determine if p(x) = 9 - 17x + x^2 belong to the span of S {4-x+3x62, 2+5x+x^2}. If it does, express one vector as a linear combination of others. otherwise , justify your answer .

 

[John Creighto]

hey
i would greatly appreciate the solutions for the question below

1) determine if p(x) = 9 - 17x + x^2 belong to the span of S {4-x+3x62, 2+5x+x^2}. If it does, express one vector as a linear combination of others. otherwise , justify your answer .

If p(x) is in the span of S then p(x)=a(4-x+3x62)+b(2+5x+x^2). Equate coefficients of the polynomial and solve the linear system of equations for the unknowns a and b.

 

[Mark44]

In general, a given vector is in the span of some set of vectors is a linear combination of the vectors in the set.

If you are working with a function space (as you are) rather than a vector space, replace the word "vector" in the previous sentence with "function."

The term "linear combination" of things in a set means a sum of scalar multiples of the things in the set.