A finite-dimensional vector space is a mathematical structure that consists of vectors, which can be added together and multiplied by scalars. The key characteristic of these spaces is that they have a finite basis, meaning a limited number of vectors can be combined to express any vector in the space. For example, in a 3-dimensional space, any vector can be represented as a combination of three basis vectors. These vector spaces are commonly used in various fields, including physics, engineering, and computer science. The dimension of a vector space is determined by the number of vectors in its basis, providing a way to understand the complexity and behavior of the space.