What characterizes a single non-zero vector in terms of linear independence?

A single non-zero vector is characterized as linearly independent because it cannot be expressed as a linear combination of other vectors—in this case, there are no other vectors involved, as it is the only vector in consideration. By definition, a set of vectors is linearly independent if no vector in the set can be written as a linear combination of the others; since there are no other vectors to combine with, this condition is satisfied with just one non-zero vector.

Furthermore, the linear independence of a single non-zero vector implies that it can span a subspace, specifically a one-dimensional subspace that consists of all scalar multiples of that vector. This directly contradicts the idea of linear dependence, which would require that the vector could relate to some other vectors in a way that implies redundancy, which is not the case here.

In summary, a single non-zero vector is a fundamental building block in linear algebra, forming the basis for a one-dimensional vector space, and thus is indeed linearly independent.