Let [itex]W = \{ax^3 + bx^2 + cx + d : b + c + d = 0\}[/itex] and [itex]P_3[/itex] be the set of all polynomials of degrees 3 or less.(adsbygoogle = window.adsbygoogle || []).push({});

So say we want to prove the [itex]W[/itex] is a subspace of [itex]P_3[/itex]. We let [itex]p(x) = a_1x^3 + a_2x^2 + a_3x + a_4[/itex] and [itex]g(x) = b_1x^3 + b_2x^2 + b_3x + b_4[/itex]. So, we compute [itex]f(x) + kg(x)[/itex] and the answer should be a polynomial of third degree or less? Is this enough?

And to find a basis for the subspace, we let [itex]b = -c - d[/itex] and just replace into the polynomial and form a basis. How can we find a vector in [itex]P_3[/itex] but not in the subspace [itex]W[/itex]?

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# W a subspace of P3

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