I need to know if there is a way to solve:
v1 = a + d
v2 = a + j
v3 = g + d
v4 = g + j
With v1, v2, v3, v4 independent there are 4 unknowns. It appears the determinant is zero or sparse.
I have no idea what you mean here! Are the "four unknowns" a, d, g, and j? And , , , and are constants? But what do you mean by "independent"? Are they vectors?
Yes, row-reducing leads to a matrix with last row all 0s. That gives . If is not equal to then there is no solution. If then there are an infinite number of solutions.