4.2: Elementary properties of vector spaces
- Page ID
- 315
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)We are going to prove several important, yet simple, properties of vector spaces. From now on, \(V\) will denote a vector space over \(\mathbb{F}\).
Proposition 4.2.1. Every vector space has a unique additive identity.
Proof. Suppose there are two additive identities \(0\) and \(0'\) Then
\[ 0'=0+0'=0, \]
where the first equality holds since \(0\) is an identity and the second equality holds since \(0'\) is an identity. Hence \(0=0'\), proving that the additive identity is unique.
Proposition 4.2.2. Every \(v \in V\) has a unique additive inverse.
Proof. Suppose \(w\) and \(w'\) are additive inverses of \(v\) so that \(v+w=0\) and \(v+w'=0\) Then
\[ w = w+0 = w+(v+w') = (w+v)+w' = 0+w' =w'. \]
Hence \(w=w'\), as desired.
Since the additive inverse of \(v\) is unique, as we have just shown, it will from now on be denoted by \(-v\) We also define \(w-v\) to mean \(w+(-v)\) We will, in fact, show in Proposition 4.2.5 below that \(-v=-1 v\)
Proposition 4.2.3. \(0v = 0\) for all \(v \in V\).
Note that the \(0\) on the left-hand side in Proposition 4.2.3 is a scalar, whereas the \(0\) on the right-hand side is a vector.
Proof. For \(v\in V\), we have by distributivity that
\[ 0v=(0+0)v=0v+0v. \]
Adding the additive inverse of \(0v\) to both sides, we obtain
\[ 0 = 0v-0v = (0v+0v)-0v=0v.\]
Proposition 4.2.4. \(a0=0\) for every \( a \in \mathbb{F}\).
Proof. As in the proof of Proposition 4.2.3, if \(a\in \mathbb{F}\), then
\[ a0=a(0+0)=a0+a0. \]
Adding the additive inverse of \(a0\) to both sides, we obtain \(0=a0\), as desired.
Proposition 4.2.5. \((-1)v=-v\) for every \(v \in V\).
Proof. For \(v\in V\), we have
\[ v+(-1)v= 1v+(-1)v = (1+(-1))v=0v=0, \]
which shows that \((-1)v\) is the additive inverse \(-v\) of \(v\).
Contributors
- Isaiah Lankham, Mathematics Department at UC Davis
- Bruno Nachtergaele, Mathematics Department at UC Davis
- Anne Schilling, Mathematics Department at UC Davis
Both hardbound and softbound versions of this textbook are available online at WorldScientific.com.