# PreTeXt Sample Book: Abstract Algebra (SAMPLE ONLY)

## Exercises5.7True/False Exercises

### 1.True/False.

Every vector space has finite dimension.
• 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.
Hint.
$$P_n\text{,}$$ the vector space of polynomials with degree at most $$n\text{,}$$ has dimension $$n+1$$ by Theorem 3.2.16. [Cross-reference is just a demo, content is not relevant.] What happens if we relax the defintion and remove the parameter $$n\text{?}$$