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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.

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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