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

### 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.