If the dimension of a vector space V is n, what can be inferred about its vectors?

The correct answer is that any set of n linearly independent vectors can form a basis for a vector space V of dimension n. This concept is rooted in the definition of a basis in linear algebra, which requires that the set of vectors not only spans the vector space but is also linearly independent.

In a vector space of dimension n, any set of n vectors can potentially serve as a basis, but to qualify, these vectors must be linearly independent. Linear independence means that no vector in the set can be expressed as a linear combination of the others. If you have n linearly independent vectors in an n-dimensional space, they ensure that every vector in that space can be represented as a unique linear combination of these basis vectors.

The dimension of the space indicates the maximum number of linearly independent vectors it can accommodate. Therefore, having a set of n linearly independent vectors confirms that they span the entire vector space V, thus satisfying the conditions for forming a basis.

When considering the other options, they do not align with the properties of vector spaces. For instance, while any set of n vectors might include dependent vectors that cannot form a basis, multiple bases can exist within the same dimensional space, and a vector space can indeed contain up to