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Row Space

Module by: Doug Daniels, Steven J. Cox. E-mail the authors

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 Ra⁢AT 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., Ra⁢AT≡ AT⁢y 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 -1010 0001 ) A 0100 -1010 0001
(1)
The row space of this matrix is:
Ra⁢AT= 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 Ra⁢AT 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 Ra⁢AT 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

Ra⁢AT=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 Ra⁢AT Ra A .

With respect to our example,

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

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Definition of a lens

Lenses

A lens is a custom view of the content in the repository. You can think of it as a fancy kind of list that will let you see content through the eyes of organizations and people you trust.

What is in a lens?

Lens makers point to materials (modules and collections), creating a guide that includes their own comments and descriptive tags about the content.

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What are tags? tag icon

Tags are descriptors added by lens makers to help label content, attaching a vocabulary that is meaningful in the context of the lens.

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