A set is said to be linearly independent by definition if implies for all i.
Suppose are linearly dependent, then there exist coefficients, not all of which are 0, such that . Notice that if then we would contradict the linear independence of . Thus we can say Showing it to be in the span as desired.
Conversely if y is in the span of then there exist coefficients such that but then we would have all of which are not zero, so is linearly dependent.