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The question provided asks which of the given statements about the vector space \( V = \{(x, y, z) \ | \ x + y = 0, z = 0\} \) is false. Let's break it down:

### Understanding the Space \( V \):
1. The condition \( x + y = 0 \) implies \( y = -x \), so every vector in \( V \) can be written as \( (x, -x, 0) \).
2. Thus, \( V \) is a subspace of \( \mathbb{R}^3 \), specifically a plane where the third coordinate is always 0 and the second coordinate is the negative of the first. 
3. Every vector in \( V \) is of the form \( x(1, -1, 0) \), meaning \( V \) is spanned by the single vector \( (1, -1, 0) \).

### Let's analyze the statements:

1. **If \( \{u\} \) belongs to \( V \), then \( \{u\} \) is a basis for \( V \):**
   - This is false. For a set to be a basis of \( V \), it must both span \( V \) and be linearly independent. Since \( V \) is spanned by a single vector, \( \{u\} \) must contain that vector (or a scalar multiple of it) to form a basis. This would not be true for just any vector \( u \) unless it's specifically the spanning vector.

2. **If \( u, v \) are in \( V \), then \( \{u, v\} \) must be linearly dependent:**
   - This is true. \( V \) is a one-dimensional space, so any two vectors in \( V \) must be scalar multiples of each other, making them linearly dependent.

3. **If \( \text{span}\{u\} = V \), then \( \{u\} \) is a basis for \( V \):**
   - This is true. If a single vector \( u \) spans \( V \), it must be the basis, since a one-dimensional space requires only one vector for its basis.

4. **If \( S \) is a basis for \( V \), then \( S \) must contain exactly one vector:**
   - This is true. Since \( V \) is one-dimensional, any basis for it must contain exactly one vector.

### Conclusion:
The **first statement** is the false one because not every \( u \) in \( V \) can form a basis.

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Do you want further details or have any other questions?  
Here are 5 questions that build on this concept:

1. How do you verify if a set of vectors is linearly dependent?
2. What are the properties of a basis in a vector space?
3. How do you determine the dimension of a subspace?
4. What are the criteria for a set of vectors to span a vector space?
5. Can a subspace have more than one basis? How are they related?

**Tip:** A basis of a vector space gives the minimal set of vectors needed to describe the entire space through linear combinations.

Let V = {(x, y, z) | x + y = 0, z = 0}. Which of the following statements is false?

Learn to analyze statements about the vector space V = {(x, y, z) | x + y = 0, z = 0}, focusing on concepts such as linear dependence, vector bases, and dimension. This problem is suitable for undergraduate students studying linear algebra.

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