What is one of the outcomes of having one vector expressed as a linear combination of others in a dependent set?

When one vector is expressed as a linear combination of others in a dependent set, it confirms the dependence of the entire set. This means that the presence of that vector does not contribute any new dimensions or direction to the span of the set; instead, it relies on the other vectors for its representation. Linear dependence indicates that at least one vector can be expressed as a combination of others, reinforcing the idea that not all vectors in the set are necessary to span the same space.

This outcome is significant in linear algebra, as it leads to implications for dimensionality. The dependent nature of the set means that there are redundancies; effectively, some vectors do not add distinct information to the span of the set. Recognizing this dependence is crucial for understanding the relationships between vectors and for reducing the set to a basis, which consists of only the necessary vectors that span the same space with independence.