Let $\mathcal{P}_2$ be the vector space of all polynomials of degree 2 or less, and let $H$ be the subspace spanned by $x-2x^{2}-2, \ 23x^{2}-34x+16$ and $5x-4x^{2}-3$.

1. The dimension of the subspace $H$ is .

2. Is $\lbrace x-2x^{2}-2, 23x^{2}-34x+16, 5x-4x^{2}-3 \rbrace$ a basis for $\mathcal{P}_2$? Be sure you can explain and justify your answer.

3. A basis for the subspace $H$ is $\big\lbrace$ $\big\rbrace$. Enter a polynomial or a comma separated list of polynomials.