Exercise 3

For each, list three elements and then show it is a vector space.

• The set of linear polynomials $\mathcal{P}_{1}=\text{$\{\hskip 1.00006pta_{0}+a_{1}x\mid a_{0},a_{1}\in\mathbb{R}\hskip 1.00006pt\}$}$ under the usual polynomial addition and scalar multiplication operations.

• The set of linear polynomials $\{\hskip 1.00006pta_{0}+a_{1}x\mid a_{0}-2a_{1}=0\hskip 1.00006pt\}$, under the usual polynomial addition and scalar multiplication operations.

Hint. Use Example 1.3 as a guide. Most of the ten conditions are just verifications.