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PreTeXt Sample Book: Abstract Algebra (SAMPLE ONLY)

Exercises 5.8 True/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,x2,x3,}, which is infinte.
  • False.

  • The vector space of all polynomials with finite degree has a basis, B={1,x,x2,x3,}, which is infinte.
Hint.
Pn, 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?