We say that a set of vectors {

$v_1, \cdots , v_p$} in

$\Bbb{R}^n$ is

**linearly independent** if:

$v_1 x_1+v_2 x_2+\cdots+v_p x_p=0$
gives only the trivial solution. In other words, the only solution is:

We say that a set of vectors {

$v_1, \cdots , v_p$} in

$\Bbb{R}^n$ is

**linearly dependent** if:

$v_1 x_1+v_2 x_2+\cdots+v_p x_p=0$
gives a non-trivial solution. In other words, they are

**linearly dependent** if it has a general solution (aka has free variable).

We can determine if the vectors is

**linearly independent** by combining all the columns in matrix (denoted as A) and solving for

$Ax=0$
__Fast way to tell if 2 or more vector are linearly dependent__
1. The vectors are multiples of one another

2. There are more vectors than there are entries in each vector.

3. There is a zero vector