Let V be the vector space

Question

Let V be the vector space P₃ [x] of polynomials in a with degree less than 3 and W be the subspace W = span{x - 5 + 4x², 2x² − (2 + 3x)}.

Find a nonzero polynomial p(x) in W. p(x) =

Find a polynomial g(x) in V \ W. q(x) =

Answer 1

Match terms to vector components:

(x - 5 + 4x^2) ~ (-5, 1, 4) and (2x^2 - (2 + 3x)) ~ (-2, -3, 2)

Take the cross product of the two vectors:

(-5, 1, 4) cross (-2, -3, 2) = (14, 2, 17)

Reproduce the polynomial in V \ W:

g(x) = 14 + 2x + 17x^2