Row Space 2.6 2001/08/09 2002/08/02 00:00:00.006 GMT-5 Doug Daniels rainking@alumni.rice.edu Steven Cox cox@rice.edu Doug Daniels rainking@alumni.rice.edu Mark Barrett markb@alumni.rice.edu Steven Cox cox@rice.edu rowspace columnspace column space row column space fundamental theorem linear algebra This module defines precisely what a row space is, gives an example of one, and then a method for finding one given an arbitrary matrix.

The Row Space As the columns of A are simply the rows of A we call Ra A the row space of A . More precisely Row Space The row space of the m-by-n matrix A is simply the span of its rows, i.e., Ra A A y y ℝ m This is a subspace of ℝ n .

Example Let us examine the matrix: A 0100 -1010 0001 The row space of this matrix is: Ra A y ℝ 3 y 1 0100 y 2 -1010 y 3 0001 As these three rows are linearly independent we may go no further. We "recognize" then Ra A as a three dimensional subspace of ℝ 4 .

Method for Finding the Basis of the Row Space Regarding a basis for Ra A we recall that the rows of A red , the row reduced form of the matrix A , are merely linear combinations of the rows of A and hence Ra A Ra A red This leads immediately to: A Basis for the Row Space Suppose A is m-by-n. The pivot rows of A red constitute a basis for Ra A . With respect to our example, 0100 -1010 0001 comprises a basis for Ra A .