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Row Space

Module by: Doug Daniels, Steven Cox

Summary: 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 AT A are simply the rows of AA we call RaAT Ra A the row space of AT A . More precisely

Definition 1: Row Space
The row space of the m-by-n matrix A A is simply the span of its rows, i.e., RaAT≡ {ATy|y∈ ℝ m } Ra A A y y ℝ m
This is a subspace of ℝ n ℝ n .

Example

Let us examine the matrix:

A=0100-10100001 A 0100 -1010 0001 (1)
The row space of this matrix is:
RaAT={ y 1 0100+ y 2 -1010+ y 3 0001|y∈ ℝ 3 } Ra A y ℝ 3 y 1 0100 y 2 -1010 y 3 0001 (2)

As these three rows are linearly independent we may go no further. We "recognize" then RaAT Ra A as a three dimensional subspace of ℝ 4 ℝ 4 .

Method for Finding the Basis of the Row Space

Regarding a basis for RaAT Ra A we recall that the rows of A red A red , the row reduced form of the matrix A A, are merely linear combinations of the rows of A A and hence

RaAT=Ra A red Ra A Ra A red (3)
This leads immediately to:

Definition 2: A Basis for the Row Space
Suppose A A is m-by-n. The pivot rows of A red A red constitute a basis for RaAT Ra A .

With respect to our example,

0100-10100001 0100 -1010 0001 (4)
comprises a basis for RaAT Ra A .

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