i'm a bit iffy about the geometric significance of linear dependence/independence myself. however, all you really need to know is that the vectors a, b and c are linearly dependent if there exists non-zero constants p, q, r such that pa + qb + rc = 0 (the letters don't matter, all you need to know is the idea). if a set of vectors aren't linearly dependent, then they're linearly independent. the a = mb + mc, etc. are merely 'shortcuts' that you can take.
the explanation below given by wikipedia represents my current understanding of linear dependence:
"A geographic example may help to clarify the concept of linear independence. A person describing the location of a certain place might say, "It is 5 miles north and 6 miles east of here." This is sufficient information to describe the location, because the geographic coordinate system may be considered as a 2-dimensional vector space (ignoring altitude). The person might add, "The place is 7.81 miles northeast of here." Although this last statement is true, it is not necessary.
In this example the "5 miles north" vector and the "6 miles east" vector are linearly independent. That is to say, the north vector cannot be described in terms of the east vector, and vice versa. The third "7.81 miles northeast" vector is a linear combination of the other two vectors, and it makes the set of vectors linearly dependent, that is, one of the three vectors is unnecessary.
Also note that if altitude is not ignored, it becomes necessary to add a third vector to the linearly independent set. In general, n linearly independent vectors are required to describe any location in n-dimensional space."