Question

Show that any set of functions containing the function identically zero is linearly dependent.

Answer 1

If a set S =[v1,...,vp] in R^n contains the zero vector, the the set is linearly dependent.
By renumbering the vectors, we may assume that v1=0 and the equation 1v1+0v2+...+0vp=0 shows that S is linearly dependent.

Answer 2

The number of vectors doesn't exceed the number of entries in each vector and since there is a zero vector in a set, it's linearly dependent