10: Linear Independence
- Page ID
- 1732
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Consider a plane \(P\) that includes the origin in \(\Re^{3}\) and a collection \(\{u,v,w\}\) of non-zero vectors in \(P\):
If no two of \(u, v\) and \(w\) are parallel, then \(P= span \{u,v,w\}\). But any two vectors determines a plane, so we should be able to span the plane using only two of the vectors \(u,v,w\). Then we could choose two of the vectors in \(\{u,v,w\}\) whose span is \(P\), and express the other as a linear combination of those two. Suppose \(u\) and \(v\) span \(P\). Then there exist constants \(d^{1}, d^{2}\) (not both zero) such that \(w=d^{1}u+d^{2}v\). Since \(w\) can be expressed in terms of \(u\) and \(v\) we say that it is not independent. More generally, the relationship
\[c^{1}u+c^{2}v+c^{3}w=0 \qquad c^{i} \in \Re, \textit{ some \(c^{i}\neq 0\)}\]
expresses the fact that \(u,v,w\) are not all independent.
Definition (Independent)
We say that the vectors \(v_{1}, v_{2}, \ldots, v_{n}\) are \(\textit{linearly dependent}\) if there exist constants (usually our vector spaces are defined over \(\mathbb{R}\), but in general we can have vector spaces defined over different base fields such as \(\mathbb{C}\) or \(\mathbb{Z}_{2}\). The coefficients \(c^{i}\) should come from whatever our base field is (usually \(\mathbb{R}\)).} \(c^{1}, c^{2}, \ldots, c^{n}\) not all zero such that
\[c^{1}v_{1} + c^{2}v_{2}+ \cdots +c^{n}v_{n}=0.\]
Otherwise, the vectors \(v_{1}, v_{2}, \ldots, v_{n}\) are \(\textit{linearly independent.}\)
Remark
The zero vector \(0_{V}\) can \(\textit{never}\) be on a list of independent vectors because \(\alpha 0_{V}=0_{V}\) for any scalar \(\alpha\).
Example \(\PageIndex{1}\):
Consider the following vectors in \(\Re^{3}\):
\[
v_{1}=\begin{pmatrix}4\\-1\\3\end{pmatrix}, \qquad
v_{2}=\begin{pmatrix}-3\\7\\4\end{pmatrix}, \qquad
v_{3}=\begin{pmatrix}5\\12\\17\end{pmatrix}, \qquad
v_{4}=\begin{pmatrix}-1\\1\\0\end{pmatrix}.
\]
Are these vectors linearly independent?
No, since \(3v_{1}+2v_{2}-v_{3}+v_{4}=0\), the vectors are linearly \(\textit{dependent}\).
Contributor
David Cherney, Tom Denton, and Andrew Waldron (UC Davis)