In a set of linearly dependent vectors, at least one vector can be expressed as what?

In a set of linearly dependent vectors, at least one vector can be expressed as a linear combination of the others. This is the defining characteristic of linear dependence. When we say that a set of vectors is linearly dependent, it means that there exists at least one vector in the set that can be represented as a sum of scalar multiples of the other vectors in that set.

For example, if you have vectors v1, v2, and v3, and they are linearly dependent, you could find scalars a and b such that one of the vectors, say v3, can be written as v3 = av1 + bv2. This representation illustrates the essence of linear dependence, where the inclusion of one vector does not introduce any new direction in the vector space spanned by the others.

The other options, while they may seem plausible, do not capture the essence of linear dependence as accurately. A random scalar multiple doesn’t guarantee that the vector can be formulated in relation to the others comprehensively. Polynomial functions speak to a different kind of relationship than what linear dependence signifies, and a vector that does not relate to the others indicates independence, which is the opposite of linear dependence. Thus, the correct understanding is that