If V is an n-dimensional vector space and if S is a set in V with exactly n vectors, then S is a basis for V if either S spans V or S is linearly independent. and If V is an n-dimensional vector space and if S is a set in V with exactly n vectors, then S is a basis for V if either S spans V or S is linearly independent.

The statement you have provided involves two important conditions that characterize a basis of a vector space, particularly when the number of vectors in the set \( S \) matches the dimension of the vector space \( V \). Let's break down both statements:

### 1. If \( V \) is an \( n \)-dimensional vector space and \( S \) is a set in \( V \) with exactly \( n \) vectors, then \( S \) is a basis for \( V \) if **either** \( S \) spans \( V \) or \( S \) is linearly independent.

- **Explanation**:
  - A **basis** of a vector space \( V \) is a set of vectors that both spans \( V \) and is linearly independent.
  - If \( S \) contains exactly \( n \) vectors (the same number as the dimension of \( V \)) and **either**:
    - \( S \) spans \( V \) (meaning that every vector in \( V \) can be written as a linear combination of the vectors in \( S \)), or
    - \( S \) is linearly independent (meaning that none of the vectors in \( S \) can be written as a linear combination of the others),
    - then it automatically satisfies the other condition. In other words, if \( S \) spans \( V \), it must also be linearly independent, and vice versa.

This fact arises from the fundamental theorem of finite-dimensional vector spaces:
- In an \( n \)-dimensional vector space, any set of \( n \) vectors that spans the space must also be linearly independent, and any set of \( n \) linearly independent vectors must span the space.

### 2. If \( V \) is an \( n \)-dimensional vector space and \( S \) is a set in \( V \) with exactly \( n \) vectors, then \( S \) is a basis for \( V \) if **either** \( S \) spans \( V \) or \( S \) is linearly independent.

- This is essentially a repeat of the first statement and emphasizes the same point.

### Conclusion:
- If \( S \) has \( n \) vectors (where \( n \) is the dimension of \( V \)) and either spans \( V \) or is linearly independent, then \( S \) is a basis for \( V \).

Would you like more details on any specific part of this concept?

### Related Questions:
1. What does it mean for a set to span a vector space?
2. How can you check if a set of vectors is linearly independent?
3. Can a set of more than \( n \) vectors in an \( n \)-dimensional space ever be linearly independent?
4. How does the dimension of a vector space relate to its basis?
5. Can a vector space have multiple bases?

### Tip:
When determining whether a set of vectors forms a basis, always compare the number of vectors in the set with the dimension of the space. If they match, you only need to verify either linear independence or spanning.

If V is an n-dimensional vector space and if S is a set in V with exactly n vectors, then S is a basis for V if either S spans V or S is linearly independent.

Explore the conditions under which a set of vectors forms a basis for an n-dimensional vector space. This topic covers key linear algebra concepts like spanning sets, linear independence, and the fundamental theorem of finite-dimensional vector spaces. Suitable for undergraduate students studying linear algebra.

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