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Let U be the subspace of P2 consisting of quadratics for which f(-6)=f(-4). (a) For a...

Let U be the subspace of P2 consisting of quadratics for which f(-6)=f(-4). (a) For a generic quadratic f(x)=ax2+bx+c, write an equation for f(-6)=f(-4) and simplify it to give a constraint on the coefficients a, b and c.

(b) Treat this equation as an underdetermined linear system in a, b and c: identify free and bound variables, and from the solution identify a basis for U.

(c) Does adding the function f(x)=x to the basis in (b) create a basis for all of P2? Prove your answer.

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