Let $\mathcal{P}_3$ be the vector space of all polynomials of degree $3$ or less in the variable $x$. Let $$\begin{array}{lcl} p_1(x) & = & -1+x-x^{2}+x^{3}, \\ p_2(x) & = & -1+x-x^{2}+x^{3}, \\ p_3(x) & = & -1+2x-x^{2}+x^{3}, \\ p_4(x) & = & -3+6x-4x^{2}+4x^{3} \\ \end{array}$$ and let $\mathcal{C} = \lbrace p_1(x), p_2(x), p_3(x), p_4(x) \rbrace.$

1. Use coordinate representations with respect to the basis $\mathcal{B} = \lbrace 1, x, x^2, x^3 \rbrace$ to determine whether the set $\mathcal{C}$ forms a basis for $\mathcal{P}_3$. choose basis for P_3 not a basis for P_3
   
   
2. Find a basis for $\mathrm{span}(\mathcal{C})$. Enter a polynomial or a comma separated list of polynomials.
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3. The dimension of $\mathrm{span}(\mathcal{C})$ is .