Let V be a five-dimensional vector space, and S be a subset of V which is a basis for V, then which of the following is FALSE?

Question 8 options:

S can have any number of elements except zero

S must have exactly five elements

S must span V

S must contain of at least five elements

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**.

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**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.

Let V be a five-dimensional vector space, and S be a subset of V which is a basis for V, then which of the following is FALSE? Question 8 options: S can have any number of elements except zero, S must have exactly five elements, S must span V, S must contain at least five elements.

Explore the properties of a basis in a five-dimensional vector space, focusing on linear independence and spanning. This problem dissects key concepts in linear algebra and determines which statements about the basis of the vector space are false. Perfect for university students studying linear algebra.

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