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The question asks to identify which of the statements is **FALSE** regarding subspaces. Let's analyze each option:

1. **Every subspace contains infinitely many vectors**:
   - This is false because the trivial subspace, which only contains the zero vector, is a valid subspace but contains only one vector. Therefore, not every subspace has infinitely many vectors.

2. **Every subspace must contain the zero vector**:
   - This is true. One of the defining properties of a subspace is that it must contain the zero vector.

3. **The set containing only the zero vector is a subspace**:
   - This is true. The set \(\{0\}\), which contains only the zero vector, is a valid subspace called the "trivial subspace."

4. **Every nonzero subspace must contain a line**:
   - This is true. Any nonzero subspace in a vector space contains at least one non-zero vector, and since subspaces are closed under scalar multiplication, any nonzero subspace contains a line through the origin.

### Conclusion:
The **first statement** ("Every subspace contains infinitely many vectors") is **false**. The correct answer should be the first option, not the third.

Would you like more details or have any questions? 

Here are 5 related questions:
1. What are the conditions for a subset of a vector space to be a subspace?
2. Can a subspace consist of only the zero vector?
3. How is scalar multiplication related to the properties of a subspace?
4. What is the geometric interpretation of a subspace containing a line?
5. How do finite-dimensional vector spaces affect the concept of infinite vectors?

**Tip**: Remember that a subspace must satisfy three properties: containing the zero vector, closure under addition, and closure under scalar multiplication.

Which of the following statement is FALSE?

1. Every subspace contains infinitely many vectors.
2. Every subspace must contain the zero vector.
3. The set containing only the zero vector is a subspace.
4. Every nonzero subspace must contain a line.

This question explores the properties of subspaces in linear algebra. It asks to identify the false statement among four options, focusing on the number of vectors in a subspace, the presence of the zero vector, and nonzero subspaces.

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