Which of the following statements about vector spaces is true?

The statement regarding vector spaces that is true is that each vector space can have multiple bases, all of the same dimension. This principle is fundamental to the concept of vector spaces in linear algebra.

A basis of a vector space is defined as a set of vectors that are linearly independent and span the space. Importantly, while a given vector space has a dimension that is defined by the number of vectors in any basis, it is possible to have different sets of vectors that serve as bases for that vector space, provided they maintain the properties of linear independence and spanning. For instance, in a 3-dimensional space, one might have a basis consisting of standard unit vectors, while another basis may be formed by linear combinations of those vectors. Despite differences in composition, all bases will contain the same number of vectors, corresponding to the dimension of the space.

In contrast, the other statements are not accurate within the context of vector spaces. Vector spaces are not confined to three dimensions; they can exist in any finite or infinite dimension. Not all vector spaces must be finite-dimensional; there are infinite-dimensional spaces, like function spaces. Additionally, while orthogonality is a property that can exist between vectors (specifically in inner product spaces), not every vector in