# OpenStax_CNX

You are here: Home » Content » Null Space

### Lenses

What is a lens?

#### 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.

##### Who can create a lens?

Any individual member, a community, or a respected organization.

##### What are tags?

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

#### Affiliated with (What does "Affiliated with" mean?)

This content is either by members of the organizations listed or about topics related to the organizations listed. Click each link to see a list of all content affiliated with the organization.
• Rice Digital Scholarship

This module is included in aLens by: Digital Scholarship at Rice UniversityAs a part of collection: "Matrix Analysis"

Click the "Rice Digital Scholarship" link to see all content affiliated with them.

#### Also in these lenses

• Lens for Engineering

This module is included inLens: Lens for Engineering
By: Sidney Burrus

Click the "Lens for Engineering" link to see all content selected in this lens.

### Recently Viewed

This feature requires Javascript to be enabled.

# Null Space

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

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

## Null Space

Definition 1: Null Space
The null space of an m-by-n matrix A A is the collection of those vectors in n n that A A maps to the zero vector in m m . More precisely, 𝒩A= x n Ax=0 𝒩 A A x 0 x n

## Null Space Example

As an example, we examine the matrix A A:

A=( 0100 -1010 0001 ) A 0100 -1010 0001
(1)

It is fairly easy to see that the null space of this matrix is:

𝒩A= t( 1 0 1 0 ) tR 𝒩 A t t 1 0 1 0
(2)

This is a line in 4 4 .

The null space answers the question of uniqueness of solutions to Sx=f S x f . For, if Sx=f S x f and Sy=f S y f then S(xy)=SxSy=ff=0 S x y S x S y f f 0 and so (xy)𝒩S x y 𝒩 S . Hence, a solution to Sx=f S x f will be unique if, and only if, 𝒩S=0 𝒩 S 0 .

## Method for Finding the Basis

Let us now exhibit a basis for the null space of an arbitrary matrix A A. We note that to solve Ax=0 A x 0 is to solve A red x=0 A red x 0 . With respect to the latter, we suppose that

c j j=1r j 1 r c j
(3)
are the indices of the pivot columns and that
c j j=r+1n j r 1 n c j
(4)
are the indices of the nonpivot columns. We accordingly define the rr pivot variables
x c j j=1r j 1 r x c j
(5)
and the nr n r free variables
x c j j=r+1n j r 1 n x c j
(6)

One solves A red x=0 A red x 0 by expressing each of the pivot variables in terms of the nonpivot, or free, variables. In the example above, x 1 x 1 , x 2 x 2 , and x 4 x 4 are pivot while x 3 x 3 is free. Solving for the pivot in terms of the free, we find x 4 =0 x 4 0 , x 3 = x 1 x 3 x 1 , x 2 =0 x 2 0 , or, written as a vector,

x= x 3 1010 x x 3 1010
(7)
where x 3 x 3 is free. As x 3 x 3 ranges over all real numbers the x above traces out a line in 4 4 . This line is precisely the null space of AA. Abstracting these calculations we arrive at:

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

## A MATLAB Observation

As usual, MATLAB has a way to make our lives simpler. If you have defined a matrix A and want to find a basis for its null space, simply call the function null(A). One small note about this function: if one adds an extra flag, 'r', as in null(A, 'r'), then the basis is displayed "rationally" as opposed to purely mathematically. The MATLAB help pages define the difference between the two modes as the rational mode being useful pedagogically and the mathematical mode of more value (gasp!) mathematically.

## Final thoughts on null spaces

There is a great deal more to finding null spaces; enough, in fact, to warrant another module. One important aspect and use of null spaces is their ability to inform us about the uniqueness of solutions. If we use the column space to determine the existence of a solution xx to the equation Ax=b A x b . Once we know that a solution exists it is a perfectly reasonable question to want to know whether or not this solution is the only solution to this problem. The hard and fast rule is that a solution xx is unique if and only if the null space of AA is empty. One way to think about this is to consider that if Ax=0 A x 0 does not have a unique solution then, by linearity, neither does Ax=b A x b . Conversely, if (Az=0)(z0)(Ay=b) A z 0 z 0 A y b then A(z+y)=b A z y b as well.

## Content actions

PDF | EPUB (?)

### What is an EPUB file?

EPUB is an electronic book format that can be read on a variety of mobile devices.

My Favorites (?)

'My Favorites' is a special kind of lens which you can use to bookmark modules and collections. 'My Favorites' can only be seen by you, and collections saved in 'My Favorites' can remember the last module you were on. You need an account to use 'My Favorites'.

| A lens I own (?)

#### 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.

##### Who can create a lens?

Any individual member, a community, or a respected organization.

##### What are tags?

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

| External bookmarks