Let's consider an arbitrary 2x2 matrix as an example,
The columns of
are linearly independent if and only if the column vectors 
are linearly independent.
This is the case if the only way we can make a linear combination of reduce to the zero vector is to multiply the vectors by 0; that is,
only by letting
.
A more concrete example: suppose
Here,
and 
. Notice that we can get the zero vector by taking 
and 
:
so the columns of are not linearly independent, or linearly dependent.
The columns of
This is the case if the only way we can make a linear combination of
only by letting
A more concrete example: suppose
Here,
so the columns of
