a family of vectors is linearly independent if none of them can be written as a linear combination of finitely many other vectors in the collection. A family of vectors which is not linearly independent is called linearly dependent. For instance, we have the following example.

v₁= k

v₂= 2j - 2k

v₃= i -  2j - k

v₄= 4i + 2j + 3k

Here the first three vectors are linearly independent; but the fourth vector equals 9 times the first plus 5 times the second plus 4 times the third, so the four vectors together are linearly dependent. Linear dependence is a property of the family, not of any particular vector; here we could just as well write the first vector as a linear combination of the last three.

a set of vectors R{v1,v2,.....vn} is said to be linearly independent if

          a1v1+ a2v2 ..........anvn=0  implies that a1=a2=......an=0

this is the mathematical definition of linearly independent vectors.

  

          it simply means that any vector belonging to that set cannot be represented by any other vector belonging to that set

            you know that any vector a can be written as

            a=tb where b is a vector parallel to a and t is an arbitrary scalar

 also a can also be represented by the linear combination of several vectors

            a= t1x + t2y+ t3z and so on.

         when this condition dosent hold for any value of t1 t2 and t3

      then vector a is said to be linearly independent of the three vectors.

 

example i , j , k are linearly independent vectors.

             i = pj+ tk can never hold good for any value of p and k.

therefore {i,j,k} is a set of linearly independent vectors.