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Exercises 3.7 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, \(B = \{1,x,x^2,x^3,\dots\}\text{,}\) 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 1.2.16 . [Cross-reference is just a demo, content is not relevant.] What happens if we relax the defintion and remove the parameter
\(n\text{?}\)
