Math  /  Algebra

QuestionLet P2\mathcal{P}_{2} be the vector space of all polynomials of degree 2 or less, and let HH be the subspace spanned by (2x+1),x22x3-(2 x+1), x^{2}-2 x-3 and 6x4x2+116 x-4 x^{2}+11 a. The dimension of the subspace HH is \square b. Is {(2x+1),x22x3,6x4x2+11}\left\{-(2 x+1), x^{2}-2 x-3,6 x-4 x^{2}+11\right\} a basis for P2\mathcal{P}_{2} ? choose \square Be sure you can explain and justify your answer. c. A basis for the subspace HH is \{ \square \}. Enter a polynomial or a comma separated list of polynomials.

Studdy Solution
A basis for the subspace HH is the linearly independent set found in Step 5:
{(2x+1),x22x3}\{-(2x+1), x^2-2x-3\}
The dimension of the subspace HH is 2\boxed{2}.
The set {(2x+1),x22x3,6x4x2+11}\{-(2x+1), x^2-2x-3, 6x-4x^2+11\} is not a basis for P2\mathcal{P}_{2}, so the answer is no\boxed{\text{no}}.
A basis for the subspace HH is {(2x+1),x22x3}\boxed{\{-(2x+1), x^2-2x-3\}}.

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