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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.

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    $\begingroup$ There is no difference between the two statements $\endgroup$ – John Glenn Mar 23 '18 at 13:22
  • $\begingroup$ In the Cartesian plane, any set of three or more vectors is linearly dependent. In order to have $n$ linearly independent vectors they must live in an $n$-dimensional space. $\endgroup$ – David K Mar 23 '18 at 13:40
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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.

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