Which two properties are required for a set of vectors to be considered a basis?

For a set of vectors to be considered a basis of a vector space, it must satisfy two essential properties: spanning the space and being linearly independent.

When a set spans a vector space, it means that any vector in the space can be expressed as a linear combination of the vectors in the set. This ensures that the entire space is covered by the span of the basis vectors.

Linear independence asserts that no vector in the set can be written as a linear combination of the others. This property guarantees that the vectors in the set contribute uniquely to the construction of other vectors in the space, ensuring that there are no redundant vectors.

Together, these properties ensure that the basis provides both coverage of the entire vector space and a unique way to represent every vector in that space, which are fundamental characteristics of a basis in linear algebra.