My goal is to find the basis vectors, and write the other vectors as a linear combination of the basis vectors.
Attached is the original matrix and the reduced matrix
I know that columns 1,2,4 are the basis vectors since they have leading ones. I tried writing the 3 basis vectors as {V1,V2,V4} | {V3,V5,V6} and solving for leading ones on the RHS but I don't think that is correct.
Since I am looking for the linear combination of V3,V5, and V6 from the 3 basis vectors I know that I need to write out something along the lines of
V3 = c1V1+c2V2+c4V4 which will give me V3.
I would prefer a bit of guidance in the right direction and not the answer.
To find $V_3$, look first at the third entry. It tells you that $c_4 = 0$, so that gives you $V_3 = c_1V_1 + c_2V_2$. Now look at the second entry: necessarily $c_2 = 2$, and now it is easy to see that $c_1 = 2$, so that $V_3 = 2V_1 + 2V_2$, as you can check.
I tried to give you step-by-step solution to make it very clear...