We want to find a basis for the row space of

(1)

We know the row space of is the subspace of spanned by the row vectors of . We also know that row-equivalent matrices have the same row space. Of course, this leads us to the fact that when we reduce to row-echelon form, say matrix , the non-zero row vectors of form a basis for the row space of Therefore,

(2)

implies the set is a basis for the row space of Note that this basis has elements, the rank of is equal to

Suppose we are given three vectors . We are interested to find a basis for the subspace of which is spanned by this set. Use the members of the given set to define a matrix as followed

(3)

Now reducing and collecting the nonzero rows will result in a basis for the row space of (which is a subspace of ). We write

(4)

The set with one member is a basis for the row space of and so it is a basis for the subspace of which is spanned by the original given set. Please note we also know that rank of is equal to since the row space of has dimension .

Suppose we are given the matrix

(5)

We know the set of all solutions of the homogeneous system of linear equations

(6)

is a subspace of In fact, this set os solutions is called the nullspace of Reducing the matrix results in

(7)

and therefore we get

(8)

Set and get

(9)

It is clear that any solution space of the given homogeneous system looks like

(10)

for any real The vector also provides a basis for the nullspace of Note that dimension of the nullspace, that is nullity, is equal to