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.