Math Vector algebra Linear dependence and independence

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:

$r\cdot\vec{a}+s\cdot\vec{b}+t\cdot\vec{c}=\vec{0}$
!

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:
$\vec{0}=\begin{pmatrix}0\\0\\0\end{pmatrix}$
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Method

1. Insert and scalar multiplication
2. Set up the equation system and solve it
3. Interpret result

Linear independence

Accordingly, three vectors are linearly independent if all coefficients are zero:

$r\cdot\vec{a}+s\cdot\vec{b}+t\cdot\vec{c}=\vec{0}$
$r=s=t=0$

Example

$\vec{a}=\begin{pmatrix}1\\1\\1\end{pmatrix}$, $\vec{b}=\begin{pmatrix}0\\2\\1\end{pmatrix}$, $\vec{c}=\begin{pmatrix}0\\1\\0\end{pmatrix}$

1. Insert and scalar multiplication

$r\cdot\vec{a}+s\cdot\vec{b}+t\cdot\vec{c}=\vec{0}$

$r\cdot\begin{pmatrix}1\\1\\1\end{pmatrix}+$ $s\cdot\begin{pmatrix}0\\2\\1\end{pmatrix}+$ $t\cdot\begin{pmatrix}0\\1\\0\end{pmatrix}$ $=\begin{pmatrix}0\\0\\0\end{pmatrix}$

$\begin{pmatrix}r\\r\\r\end{pmatrix}+$ $\begin{pmatrix}0\\2s\\s\end{pmatrix}+$ $\begin{pmatrix}0\\t\\0\end{pmatrix}$ $=\begin{pmatrix}0\\0\\0\end{pmatrix}$
2. Set up equation system and solve it

Write each line as an equation.
1. $r=0$
2. $r+2s+t=0$
3. $r+s=0$

Insert $r=0$
1. $2s+t=0$
2. $s=0$

Insert $s=0$
$t=0$
3. Interpret result

$r=0$; $s=0$; $t=0$

=>The vectors are linearly independent.