The terms span, spanning set, and basis set are often a source of confusion for students. We already understand the span of a vector set from the previous subsection. A spanning set of a subspace 
is simply any set of vectors 
for which 
. There are many ways of saying this that might appear in various textbooks:
is simply any set of vectors for which . There are many ways of saying this that might appear in various textbooks: - The span of
is 
. - The vector set
spans 
. - The vector set
is a spanning set for 
.
Can we tell from inspection whether or not a set of vectors spans a particular subspace? Let's consider two vector sets
and 
:
and 
are composed of two vectors. But don't be tricked into thinking that 
and 
both span planes. In 
, the second vector is a multiple of the first (
). In 
it is impossible to find a value 
for which 
. Thus, we say that the vectors in 
are linearly independent. Formally, a set of vectors is linearly independent if none of them can be written as a linear combination of finitely many other vectors in the set. And, the dimension of the subspace spanned by a set of vectors is equal to the number of linearly independent vectors in that set. So, 
and 
which means that 
spans a line and 
spans a plane.The discussion of linear independence leads us to the concept of a basis set. A basis is a way of specifing a subspace with the minimum number of required vectors. If
is a basis set for a subspace 
, then every vector in 
(
) can be written as 
. Moreover, the series of scalars 
is known as the coordinates of a vector relative to the basis 
.We are already very familiar with a basis and coordinate set known as the standard basis set. In 3-dimensional space this is simply the
, 
, and 
axes which are written as vectors known as the standard basis vectors for 
:
dimensions, the standard basis set is 
. So to summarize, a basis can be quite useful for defining not only a subspace within 
, but for specifying any point within that subspace with a standardized reference system called coordinates.The next question one might ask is how to determine the dimension of the span of a vector set (
) and how to find a basis set given a spanning set. To answer the first question we recall the definition of the rank of a matrix as the number of pivotal columns in the matrix. With this definition, we can gather the vectors in 
into a matrix 
and state the following:
is given by the number of pivotal columns in 
. Moreover, the columns that contain pivots in the RREF matrix correspond to the columns that are linearly independent vectors from the original matrix 
. The linear independent vectors make up the basis set. Gauss-Jordan Elimination already provides a standard algorithm for finding 
and thus the basis for 
. Let's take the following set as an example:
is a plane in 
. It is also important to note that the RREF form of the matrix gives us more information than just which vectors are linearly independent. The non-pivotal columns tell us exactly what linear combinations of the linearly independent vectors are required to give the dependent columns. For example, based on the third and fourth columns of the RREF matrix above we can say that 
and 
.Finally, you might have noticed by now that a basis is not unique for a particular subspace. If we had arranged the columns of the matrix
differently, we would have obtained a different basis.