To solve this problem, let's analyze each of the statements about a basis for a five-dimensional vector space $V$.

Key Concept: Basis of a Vector Space

A basis of a vector space is a subset of vectors that:

1. Spans the vector space, meaning every vector in the vector space can be written as a linear combination of the basis vectors.
2. Is linearly independent, meaning no vector in the basis can be written as a linear combination of the others.

For a vector space $V$ of dimension $n$, any basis must have exactly $n$ vectors.

Since $V$ is a five-dimensional vector space, a basis of $V$ must have exactly 5 vectors.

Now, let's evaluate the statements one by one:

1. S can have any number of elements except zero

• False. A basis must have exactly 5 elements, so it can't have any number of elements except zero. This implies that a set of fewer than 5 elements or more than 5 elements cannot be a basis of a five-dimensional vector space.

2. S must have exactly five elements

• True. Since $V$ is a 5-dimensional vector space, any basis of $V$ must have exactly 5 elements.

3. S must span V

• True. By definition, a basis must span the vector space.

4. S must contain at least five elements

• False. A basis must contain exactly five elements, not more than five.

Conclusion:

The statements that are FALSE are:

• "S can have any number of elements except zero" (because it must have exactly 5 elements).
• "S must contain at least five elements" (it should have exactly 5 elements).

However, since the question asks for which one is false, the first false statement is the correct answer:

• S can have any number of elements except zero is FALSE.

---

Would you like further details on any part of this explanation?

---

Relative Questions:

1. What are the properties of a linearly independent set in vector spaces?
2. How do you define the dimension of a vector space?
3. Can a set of dependent vectors span a vector space?
4. How can we determine if a given set of vectors is a basis for a space?
5. What happens if a basis has more than the required number of vectors?

Tip: In any vector space, the number of vectors in a basis is always equal to the dimension of the space.