If the set of vectors has only one trivial solution (i.e the 0 vector) of $Ax = 0$ , then they are linearly independent
If the set of vectors has non trivial solution of $Ax = 0$ , then they are linearly dependent
If the number of columns > the number of rows i.e (if a set contains more vectors than there are entries in each vector) then the set is linearly independent
which is pretty much intuitive if you think an example in 2D
of course the span of any 2 independent vector will fill the whole 2D space . Any number of vectors extra will form linearly dependent set