If B = {a1, ..., an} is a basis of V, what is true about the vectors in B?

The assertion that the vectors in the basis ( B = {a_1, \ldots, a_n} ) span the vector space ( V ) is foundational to the definition of a basis in linear algebra. A basis is defined as a set of vectors that satisfies two critical properties:

1. The vectors are linearly independent.
2. The vectors span the vector space.

The spanning property means that any vector in the vector space ( V ) can be expressed as a linear combination of the basis vectors. This allows for the representation of all vectors in ( V ) using the basis vectors, which is essential for the structure of the vector space.

Further properties stemming from this include the idea that the basis provides a minimal yet complete set of vectors needed to represent any element of the vector space. Each vector in the basis contributes uniquely to its dimensionality, reinforcing the idea that it cannot simply be made from scalar multiples of one another, nor can it be a redundant collection of vectors.

The other options reflect misunderstandings of the definition of a basis. For instance, if the vectors were all scalar multiples of each other, they wouldn't be linearly independent; if they formed a linearly dependent set, they couldn't be