Which statement is true regarding the bases of a vector space?

The statement that all bases of a vector space have the same number of vectors is rooted in the fundamental properties of vector spaces. This concept is known as the dimension of a vector space, which is defined as the number of vectors in any basis of that space.

A basis is a set of vectors that is both linearly independent and spans the vector space. Regardless of which basis you select, the number of vectors will remain constant; this is a result of the definition of dimension. Therefore, if one basis consists of ( n ) vectors, any other basis of the same vector space must also consist of ( n ) vectors. This consistency in the number of basis vectors across different bases is crucial for understanding the structure and properties of vector spaces.

In contrast, the other statements do not hold true universally. For example, while some vector spaces may have finite dimensions, there are also infinite-dimensional spaces, which makes the statement about the dimension being finite incorrect. Additionally, the dimension being always one applies only to specific cases, such as one-dimensional vector spaces, but not to all vector spaces.