For which values of the parameter a is , and a basis for ?
How can you possibly get in there? You should get a second degree polynomial in .
This works, because for the three vectors to be a basis of they should form the sides of a parallelepiped having nonzero volume, and the determinant is that volume. Or, more abstractly put, if is a basis, then the change of basis transformation from to the standard basis is invertible; its matrix representation is the one whose determinant you evaluated (with a small mistake).