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Left Null Space

Module by: Doug Daniels, Steven Cox

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.

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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∈ ℝ m |ATy=0} 𝒩 A A y 0 y ℝ 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=111213 A 11 12 13 (1)
then
AT=111123 A 111 123 (2)
and so
A red T =111012 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={y31-21|y3∈ℝ} 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=1…r} j 1 … r c j and free indices { c j |j=r+1…m} j r 1 … m c j . A basis for 𝒩AT 𝒩 A may be constructed of m−r 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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