There are certain conditions pertaining to the expression of vectors in a vector
space that allow for the individual vectors in the vector space to be expressed
as linear combinations in one particular way.  
 
         Linear Independence:  To start off, allow me to define the concept of linear 
independence:  Suppose we have the set V = {v1, v2, v3,...vr}.  If the vector 
equation, c1v1 + c2v2 + c3v3 + ... + crvr = 0 has only one solution then
(c1 = 0, c2 = 0, c3 = 0, ... cr = 0), then V is linear independent.  
If the general constant, cn, is equal to 0 or any nonzero integer, then we say that V 
is linearly dependent.  In simplistic terms, linear independence determines whether 
the trivial solution of x1 = x2 = x3 = xn = 0 is the one and only one solution.
   
       In determination of linear independence, our vectors in augmented matrix form
should be (with proper row operations) in reduced row-echelon form.