Show that a set of vectors S ⊂ X is linearly dependent if and only if there exists distinct vectors x1, x2, . . . , xn ∈ S and numbers a1, a2, . . . , an, not all zero, such that
a1x1 + a2x2 +.......+ anxn = 0 … 5†
The null vector therefore is a nontrivial linear combination of other vectors. This is an alternative characterization of linear dependence found in some texts.