Exercise 9.4.1

Let $\mathbf{P}_{3}$ be the vector space of polynomials of degree at most 3. Determine which of the following are bases for this vector space.

1. $\left\{x^{3}+1,~{}x^{2}+x,~{}2x^{3}+x^{2},~{}2x^{3}-x^{2}-3x+1\right\}$.

2. $\left\{x+1,~{}x^{3}+x^{2}+2x,~{}x^{2}+x,~{}x^{3}+x^{2}+x\right\}$.