Assume 3a, 2a-b, a+c are dependent. Then, there exist scalras , not all zero, such that

If then . Therefore not all the coefficients are zero.

Let

Then but not all of the coefficients are zero, in contradiction to the fact that a,b,c are linearly independent... ==> the assumption can not hold, therefore 3a, 2a-b and a+c are linearly independent.