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# Left Null Space

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

Summary: This module defines the left null space, shows an example of what one is, and describes how to find one given an arbitrary matrix.

## Left Null Space

If one understands the concept of a null space, the left null space is extremely easy to understand.

Definition 1: Left Null Space
The Left Null Space of a matrix is the null space of its transpose, i.e., 𝒩AT= y m ATy=0 𝒩 A A y 0 y m
The word "left" in this context stems from the fact that ATy=0 A y 0 is equivalent to yTA=0 y A 0 where yy "acts" on AA from the left.

## Example

As A red A red was the key to identifying the null space of AA, we shall see that A red T A red T is the key to the null space of AT A . If

A=( 11 12 13 ) A 11 12 13
(1)
then
AT=( 111 123 ) A 111 123
(2)
and so
A red T =( 111 012 ) A red T 111 012
(3)
We solve A red T =0 A red T 0 by recognizing that y 1 y 1 and y 2 y 2 are pivot variables while y 3 y 3 is free. Solving A red T y=0 A red T y 0 for the pivot in terms of the free we find y 2 =(2 y 3 ) y 2 2 y 3 and y 1 = y 3 y 1 y 3 hence
𝒩AT= y3( 1 -2 1 ) y3R 𝒩 A y3 y3 1 -2 1
(4)

## Finding a Basis for the Left Null Space

The procedure is no different than that used to compute the null space of AA itself. In fact

Definition 2: A Basis for the Left Null Space
Suppose that AT A is n-by-m with pivot indices c j j=1r j 1 r c j and free indices c j j=r+1m j r 1 m c j . A basis for 𝒩AT 𝒩 A may be constructed of mr m r vectors z 1 z 2 z m - r z 1 z 2 z m - r where z k z k , and only z k z k , possesses a nonzero in its c r + k c r + k component.

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