A set of vectors that is not linearly independent can lead to what kind of implications in the context of a vector space?

In the context of a vector space, a set of vectors that is not linearly independent implies that at least one of the vectors in the set can be expressed as a linear combination of the others. This situation leads to redundancy within the set of vectors, meaning that the dimension of the span of those vectors is less than the number of vectors in the set.

When discussing systems of linear equations associated with such vectors, this redundancy can result in infinite solutions. For example, if the system of equations represented by the vectors does not have full rank (due to linear dependence), there would be free variables in the corresponding solution set. As a result, one could find infinitely many solutions that satisfy the equations. This is a fundamental concept in linear algebra, emphasizing how the linear dependence among vectors creates scenarios where the solution is not unique.

The other implications offered do not hold true in the same way. A set of vectors that is not linearly independent does not guarantee a unique basis, as linear dependence undermines the uniqueness of representation in a vector space. Also, it does not directly prove finite dimension, as dimensions can be finite or infinite regardless of linear independence. Lastly, the idea of simplification of the space is not inherently linked to linear dependence; linear