A basis for a vector space is a set of linearly independent vectors whose span is the set of vectors of the space. For any vector in the space, there exists a linear combination of the basis vectors whose result is that vector. In this case, it is common to say that any vector in the space “can be written as a linear combination of the basis vectors.”

Since the basis is a linearly independent set, the linear combination of basis vectors is unique.

If we have a basis of $n$ vectors for $(V, \F )$ then each vector $v \in V$ can be written uniquely as a linear combination of the vectors in the basis. If we take the vector in the field which is these coefficients, then this is an isomorphism with the vector space $(\F ^n, \F )$ We call this the coordinate vector.

A set of vectors is a basis if and only if
no proper superset of it is linearly independent.

A set of vectors that spans the space is a
basis if and only if no proper subset of it
spans the space.