I've just go through the definition of linear independence.
Vectors $\vec x_1,\vec x_2,\ldots,\vec x_n$ are linearly independent if and only if the equation $c_1\vec x_1+c_2\vec x_2+\ldots+c_3\vec x_3=0$.
and
Vectors $\vec x_1,\vec x_2,\ldots,\vec x_n$ are linearly independent if and only if the equation $c_1\vec x_1+c_2\vec x_2+\ldots+c_3\vec x_3=0$.
It's quite easy to understand in algebraically way, but I cannot imagine in a geometry way.
How to imagine this two definitions in Cartesian plane (in a geometry way to understand)?
If there is an example to explain, it will be better.
Start from the origin, make steps in the direction of $\vec x_1$ (forward or backward, doesn't matter). Then try to come back to the origin in the direction of $\vec x_2$.
With independent vectors, it is impossible. This generalizes to higher dimensions.