SB True/False Exercises

Exercises 5.8 True/False Exercises

Every vector space has finite dimension.

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

Hint.

P_{n},

the vector space of polynomials with degree at most

n,

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 ?