What is the dimension of a vector space?

The dimension of a vector space is defined as the number of vectors in a basis for that space, which consists of a set of linearly independent vectors that span the entire vector space. This is crucial because the basis captures all the necessary directions in the space, and the dimension quantifies the minimum number of vectors needed to describe every vector in that space through linear combinations.

In contrast, while the number of vectors in the vector space or the number of vectors that can be added together may arise from the properties of a vector space, they do not accurately represent the concept of dimension, which focuses specifically on the basis and linear independence. The maximum number of linear combinations can be a misleading concept as it varies based on the chosen vectors, rather than the intrinsic structure of the vector space itself. Therefore, focusing on the count of linearly independent vectors in the basis provides a precise and meaningful measure of the dimension.