Final answer:
The only finite vector space over the rational numbers is the trivial vector space, which contains just the zero vector. This is because the rational numbers form an infinite field and any vector space over an infinite field cannot be finite. Therefore, the answer is B) 1.
Step-by-step explanation:
The question asks how many vector spaces over the rational numbers can have a finite number of elements. In the field of mathematics, and more specifically linear algebra, a vector space over a field F is a set V along with two operations that satisfy the eight axioms of vector spaces. To have a finite vector space over any field, the field itself must be finite since fields contain the scalar elements used to multiply vectors.
The rational numbers Q, however, form an infinite field. This means any vector space over Q would necessarily be infinite because you can always multiply a vector by a new rational number scalar to get a new vector in the space. Therefore, the only vector space over the rational numbers that has a finite number of elements is the trivial vector space, which consists of only the zero vector.
Following this reasoning, the correct answer to the question is B) 1, since the only finite vector space over the rational numbers is the one with just the zero vector.