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Solution
Except for a variation in notation, the space spanned by these vectors is the row space of the matrix
Reducing this matrix to row-echelon form, we obtain
The nonzero row vectors in this matrix are
These vectors form a basis for the row space and consequently form a basis for the subspace of spanned by , , , and .
Observe that in Example 6 the basis vectors obtained for the column space of A consisted of column vectors of A, but the basis
vectors obtained for the row space of A were not all row vectors of A. The following example illustrates a procedure for finding a
basis for the row space of a matrix A that consists entirely of row vectors of A.
EXAMPLE 8 Basis for the Row Space of a Matrix
Find a basis for the row space of
consisting entirely of row vectors from A.
Solution
We will transpose A, thereby converting the row space of A into the column space of ; then we will use the method of Example
6 to find a basis for the column space of ; and then we will transpose again to convert column vectors back to row vectors.
Transposing A yields
Reducing this matrix to row-echelon form yields
The first, second, and fourth columns contain the leading 1's, so the corresponding column vectors in form a basis for the
