SB True/False Exercises

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Exercises 5.7 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 3.2.16. [Cross-reference is just a demo, content is not relevant.] What happens if we relax the defintion and remove the parameter $n\text{?}$

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