10: Linear Independence

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Consider a plane $P$ that includes the origin in $\Re^{3}$ and a collection $\{u,v,w\}$ of non-zero vectors in $P$:

$151257541312468.png$

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 106

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)