If (2, -1, 2, 1) is a linear combination of the three given vectors, then there should exist such that
or equivalently, there should exist a solution to the system
Right away we get , so the system reduces to
Notice that the first equation is the sum of the latter two. The third equation gives us
so that in the second equation,
which in turn gives
and hence the (2, -1, 2, 1) is a linear combination of the given vectors, with