How to determine if the columns of a matrix are linearly independent?

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.