SB True/False Exercises

$\newcommand{\identity}{\mathrm{id}} \newcommand{\notdivide}{{\not{\mid}}} \newcommand{\notsubset}{\not\subset} \newcommand{\lcm}{\operatorname{lcm}} \newcommand{\gf}{\operatorname{GF}} \newcommand{\inn}{\operatorname{Inn}} \newcommand{\aut}{\operatorname{Aut}} \newcommand{\Hom}{\operatorname{Hom}} \newcommand{\cis}{\operatorname{cis}} \newcommand{\chr}{\operatorname{char}} \newcommand{\Null}{\operatorname{Null}} \newcommand{\lt}{<} \newcommand{\gt}{>} \newcommand{\amp}{&} \definecolor{fillinmathshade}{gray}{0.9} \newcommand{\fillinmath}[1]{\mathchoice{\colorbox{fillinmathshade}{$\displaystyle \phantom{\,#1\,}$}}{\colorbox{fillinmathshade}{$\textstyle \phantom{\,#1\,}$}}{\colorbox{fillinmathshade}{$\scriptstyle \phantom{\,#1\,}$}}{\colorbox{fillinmathshade}{$\scriptscriptstyle\phantom{\,#1\,}$}}} \newcommand{\sfrac}[2]{{#1}/{#2}} $

Exercises 3.8 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, $B = \{1,x,x^2,x^3,\dots\}\text{,}$ 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 $n\text{?}$

Prev Top Next