Linear dependence and independence
A set of vectors is linearly dependent if at least one of the vectors in the set can be defined as a linear combination of the others.
Linear dependence
To put it simply, three vectors are linearly dependent if at least one coefficient is not zero, so the following applies:
$r\cdot\vec{a}+s\cdot\vec{b}+t\cdot\vec{c}=\vec{0}$
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Remember
$\vec{0}$ is called the zero vector. This one is in the plane:
$\vec{0}=\begin{pmatrix}0\\0\end{pmatrix}$
and in space:
$\vec{0}=\begin{pmatrix}0\\0\\0\end{pmatrix}$
$\vec{0}=\begin{pmatrix}0\\0\end{pmatrix}$
and in space:
$\vec{0}=\begin{pmatrix}0\\0\\0\end{pmatrix}$
Linear independence
Accordingly, three vectors are linearly independent if all coefficients are zero:
$r\cdot\vec{a}+s\cdot\vec{b}+t\cdot\vec{c}=\vec{0}$
$r=s=t=0$
$r=s=t=0$
Example
$\vec{a}=\begin{pmatrix}1\\1\\1\end{pmatrix}$, $\vec{b}=\begin{pmatrix}0\\2\\1\end{pmatrix}$, $\vec{c}=\begin{pmatrix}0\\1\\0\end{pmatrix}$
Insert and scalar multiplication
$r\cdot\vec{a}+s\cdot\vec{b}+t\cdot\vec{c}=\vec{0}$
$r\cdot\begin{pmatrix}1\\1\\1\end{pmatrix}+$ $s\cdot\begin{pmatrix}0\\2\\1\end{pmatrix}+$ $t\cdot\begin{pmatrix}0\\1\\0\end{pmatrix}$ $=\begin{pmatrix}0\\0\\0\end{pmatrix}$
$\begin{pmatrix}r\\r\\r\end{pmatrix}+$ $\begin{pmatrix}0\\2s\\s\end{pmatrix}+$ $\begin{pmatrix}0\\t\\0\end{pmatrix}$ $=\begin{pmatrix}0\\0\\0\end{pmatrix}$Set up equation system and solve it
Write each line as an equation.- $r=0$
- $r+2s+t=0$
- $r+s=0$
Insert $r=0$- $2s+t=0$
- $s=0$
Insert $s=0$
$t=0$Interpret result
$r=0$; $s=0$; $t=0$
=>The vectors are linearly independent.