In my textbook, The vector v is a linear combination of the vectors v1, v2, and v3 if there are scalars c1, c2, and c3, such that v=c1*v1+c2*v2+c3*v3. So c has to be a "scalar." To find these c values you can set up the augmented matrix (v1, v2, v3, v) and find the RREF. I'm a little confused. If you have a free variable in in the RREF matrix, that means c1, c2, and c3 aren't "scalars." does that mean v is not a linear combination of v1, v2, v3? It would seem so from that question. However in the next section in my textbook, it shows that you can write v as a linear combination of v1, v2, v3 even if you have a free variable in the augmented matrix. Isn't that contradictory?