All the vectors of the range can be written as hence, if you manage to find a vector of which direction is not that of , you are sure it does not belong to the range of the operator.
Show that the range of the linear operator defined by the equations
w1 = 4x1 - 2x2
w2 = 2x1 - x2
is not all of R2, and find a vector that is not in the range.
Ok I can easily show the range is not all of R2 by finding the determinant to be zero... but how the vector which is not in the range? I know this is probably really easy.