Skip to main content
Contents Dark Mode Prev Up Next \(\newcommand{\nul}{\operatorname{nul}}
\newcommand{\N}{\mathbb N}
\newcommand{\Z}{\mathbb Z}
\newcommand{\Q}{\mathbb Q}
\newcommand{\C}{\mathbb C}
\newcommand{\R}{\mathbb R}
\newcommand{\avec}{{\mathbf a}}
\newcommand{\bvec}{{\mathbf b}}
\newcommand{\cvec}{{\mathbf c}}
\newcommand{\dvec}{{\mathbf d}}
\newcommand{\dtil}{\widetilde{\mathbf d}}
\newcommand{\evec}{{\mathbf e}}
\newcommand{\fvec}{{\mathbf f}}
\newcommand{\nvec}{{\mathbf n}}
\newcommand{\pvec}{{\mathbf p}}
\newcommand{\qvec}{{\mathbf q}}
\newcommand{\svec}{{\mathbf s}}
\newcommand{\tvec}{{\mathbf t}}
\newcommand{\uvec}{{\mathbf u}}
\newcommand{\vvec}{{\mathbf v}}
\newcommand{\wvec}{{\mathbf w}}
\newcommand{\xvec}{{\mathbf x}}
\newcommand{\yvec}{{\mathbf y}}
\newcommand{\zvec}{{\mathbf z}}
\newcommand{\rvec}{{\mathbf r}}
\newcommand{\mvec}{{\mathbf m}}
\newcommand{\zerovec}{{\mathbf 0}}
\newcommand{\onevec}{{\mathbf 1}}
\newcommand{\real}{{\mathbb R}}
\newcommand{\complex}{\mathbb C}
\newcommand{\field}{\mathbb F}
\newcommand{\pbb}{\mathbb P}
\newcommand{\poly}{\mathbb P}
\newcommand{\basis}[2]{#1_1,#1_2,\ldots,#1_{#2}}
\newcommand{\conjugate}[1]{\overline{#1}}
\newcommand{\conj}[1]{\overline{#1}}
\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2
\end{array}\right]}
\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2
\end{array}\right]}
\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3
\end{array}\right]}
\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3
\end{array}\right]}
\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4
\end{array}\right]}
\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4
\end{array}\right]}
\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\
#4 \\ #5 \\ \end{array}\right]}
\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\
#4 \\ #5 \\ \end{array}\right]}
\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}
\newcommand{\laspan}[1]{\operatorname{Span}\{#1\}}
\newcommand{\bcal}{{\cal B}}
\newcommand{\ccal}{{\cal C}}
\newcommand{\dcal}{{\cal D}}
\newcommand{\ecal}{{\cal E}}
\newcommand{\scal}{{\cal S}}
\newcommand{\wcal}{{\cal W}}
\newcommand{\ecal}{{\cal E}}
\newcommand{\fcal}{{\cal F}}
\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}
\newcommand{\gray}[1]{\color{gray}{#1}}
\newcommand{\lgray}[1]{\color{lightgray}{#1}}
\newcommand{\rank}{\operatorname{rank}}
\newcommand{\row}{\operatorname{Row}}
\newcommand{\col}{\operatorname{Col}}
\renewcommand{\deg}{\operatorname{deg}}
\newcommand{\range}{\operatorname{range}}
\newcommand{\tr}{\operatorname{tr}}
\newcommand{\var}{\text{Var}}
\newcommand{\corr}{\text{corr}}
\newcommand{\len}[1]{\left|#1\right|}
\newcommand{\bbar}{\overline{\bvec}}
\newcommand{\bhat}{\widehat{\bvec}}
\newcommand{\bperp}{\bvec^\perp}
\newcommand{\xhat}{\widehat{\xvec}}
\newcommand{\vhat}{\widehat{\vvec}}
\newcommand{\uhat}{\widehat{\uvec}}
\newcommand{\what}{\widehat{\wvec}}
\newcommand{\Sighat}{\widehat{\Sigma}}
\newcommand{\inner}[2]{\langle #1,#2\rangle}
\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\,}$}}}
\)
Chapter 1 Vector spaces
Our previous linear algebra course focused on vectors in \(\real^n\text{,}\) which we thought of as a list of \(n\) real numbers, such as \(\vvec=\threevec2{-1}4\text{.}\) An essential part of our study were two algebraic operations performed on vectors: scalar multiplication and vector addition.
In addition to vectors, we also considered matrices and their associated matrix transformations. At times, we also considered scalar multiplication and addition of matrices. For instance, when studying the eigenvalues and eigenvectors of a matrix \(A\text{,}\) we may have constructed the associated matrix \(A-2I\text{.}\) In other words, we multiplied the matrix \(I\) by the scalar \(-2\) and then added \(A\text{.}\)
In fact, there are other types of mathematical objects for which we can define operations like scalar multiplication and vector addition. Polynomials are a good example. If \(p(x)=2x^2
- 4\) and \(q(x)=x+7\text{,}\) then we can multiply \(p(x)\) by a scalar and add two polynomials:
\begin{align*}
3p(x)\amp=6x^2 - 12 \\
p(x)+q(x)\amp=2x^2+x+3
\end{align*}
Our first task in this course is to generalize our earlier work with vectors in \(\real^n\) and matrices to things like polynomials. This leads us to the idea of a vector space.
