#### 1. True/False.

True.

- The vector space of all polynomials with finite degree has a basis, \(B = \{1,x,x^2,x^3,\dots\}\text{,}\) which is infinte.
False.

- The vector space of all polynomials with finite degree has a basis, \(B = \{1,x,x^2,x^3,\dots\}\text{,}\) which is infinte.

Every vector space has finite dimension.

Hint.

\(P_n\text{,}\) the vector space of polynomials with degree at most \(n\text{,}\) has dimension \(n+1\) by Theorem 1.2.16. [Cross-reference is just a demo, content is not relevant.] What happens if we relax the defintion and remove the parameter \(n\text{?}\)