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Chapter 26. Null Space

The null space of a matrix is the set of all vectors that the matrix sends to zero. If AA is an m×nm \times n matrix, then its null space is the solution set of the homogeneous system

Ax=0. Ax=0.

Equivalently, the null space is the kernel of the linear transformation x↦Axx \mapsto Ax. It is always a subspace of the domain FnF^n.

26.1 Definition

Let AA be an m×nm \times n matrix over a field FF. The null space of AA, denoted

Null⁡(A), \operatorname{Null}(A),

is defined by

Null⁡(A)={x∈Fn:Ax=0}. \operatorname{Null}(A) = \{x \in F^n : Ax=0\}.

The vectors in the null space are the solutions of a homogeneous linear system.

Since AA has nn columns, the input vector xx has nn components. Therefore

Null⁡(A)⊆Fn. \operatorname{Null}(A) \subseteq F^n.

The null space lives in the domain of the matrix transformation, not in the codomain.

26.2 Homogeneous Systems

A system is homogeneous when its right-hand side is zero:

Ax=0. Ax=0.

Such a system is always consistent because

x=0 x=0

is always a solution.

This solution is called the trivial solution.

The main question is whether there are nontrivial solutions:

x≠0. x \neq 0.

Nontrivial solutions exist exactly when the columns of AA are linearly dependent.

26.3 Null Space as a Subspace

The null space is a subspace of FnF^n.

First, it contains the zero vector because

A0=0. A0=0.

Next, let

u,v∈Null⁡(A). u,v \in \operatorname{Null}(A).

Then

Au=0 Au=0

and

Av=0. Av=0.

Therefore

A(u+v)=Au+Av=0+0=0. A(u+v)=Au+Av=0+0=0.

So

u+v∈Null⁡(A). u+v \in \operatorname{Null}(A).

For a scalar cc,

A(cu)=cAu=c0=0. A(cu)=cAu=c0=0.

So

cu∈Null⁡(A). cu \in \operatorname{Null}(A).

Thus Null⁡(A)\operatorname{Null}(A) is closed under addition and scalar multiplication.

26.4 Geometric Meaning

The null space contains all directions that are collapsed to zero by the transformation

x↦Ax. x \mapsto Ax.

If

x∈Null⁡(A), x \in \operatorname{Null}(A),

then

Ax=0. Ax=0.

Thus xx is invisible to the transformation: adding xx to an input does not change the output.

Indeed, if Ax=0Ax=0, then for any vector vv,

A(v+x)=Av+Ax=Av. A(v+x)=Av+Ax=Av.

So null-space vectors are directions of ambiguity.

26.5 Example in R2\mathbb{R}^2

Let

A=[12]. A= \begin{bmatrix} 1 & 2 \end{bmatrix}.

Then

Ax=0 Ax=0

means

[12][xy]=0. \begin{bmatrix} 1 & 2 \end{bmatrix} \begin{bmatrix} x\\ y \end{bmatrix} =0.

So

x+2y=0. x+2y=0.

Solving for xx,

x=−2y. x=-2y.

Let

y=t. y=t.

Then

x=−2t. x=-2t.

Thus

[xy]=[−2tt]=t[−21]. \begin{bmatrix} x\\ y \end{bmatrix} = \begin{bmatrix} -2t\\ t \end{bmatrix} = t \begin{bmatrix} -2\\ 1 \end{bmatrix}.

Therefore

Null⁡(A)=span⁡([−21]). \operatorname{Null}(A) = \operatorname{span} \left( \begin{bmatrix} -2\\ 1 \end{bmatrix} \right).

This null space is a line through the origin in R2\mathbb{R}^2.

26.6 Example in R3\mathbb{R}^3

Let

A=[111]. A= \begin{bmatrix} 1 & 1 & 1 \end{bmatrix}.

The equation

Ax=0 Ax=0

becomes

x1+x2+x3=0. x_1+x_2+x_3=0.

Solve for x1x_1:

x1=−x2−x3. x_1=-x_2-x_3.

Let

x2=s,x3=t. x_2=s, \qquad x_3=t.

Then

x=[−s−tst]=s[−110]+t[−101]. x= \begin{bmatrix} -s-t\\ s\\ t \end{bmatrix} = s \begin{bmatrix} -1\\ 1\\ 0 \end{bmatrix} + t \begin{bmatrix} -1\\ 0\\ 1 \end{bmatrix}.

Therefore

Null⁡(A)=span⁡([−110],[−101]). \operatorname{Null}(A) = \operatorname{span} \left( \begin{bmatrix} -1\\ 1\\ 0 \end{bmatrix}, \begin{bmatrix} -1\\ 0\\ 1 \end{bmatrix} \right).

The null space is a plane through the origin in R3\mathbb{R}^3.

26.7 Finding the Null Space

To find Null⁡(A)\operatorname{Null}(A), solve

Ax=0. Ax=0.

The standard procedure is:

StepOperation
1Row reduce AA to echelon or reduced echelon form
2Identify pivot variables and free variables
3Express pivot variables in terms of free variables
4Write the solution in parametric vector form
5Extract a basis from the parameter vectors

The parameter vectors form a basis for the null space.

26.8 Row Reduction Example

Let

A=[121024311221]. A= \begin{bmatrix} 1 & 2 & 1 & 0\\ 2 & 4 & 3 & 1\\ 1 & 2 & 2 & 1 \end{bmatrix}.

Solve

Ax=0. Ax=0.

Row reduce:

[121024311221]→[121000110011]→[121000110000]. \begin{bmatrix} 1 & 2 & 1 & 0\\ 2 & 4 & 3 & 1\\ 1 & 2 & 2 & 1 \end{bmatrix} \to \begin{bmatrix} 1 & 2 & 1 & 0\\ 0 & 0 & 1 & 1\\ 0 & 0 & 1 & 1 \end{bmatrix} \to \begin{bmatrix} 1 & 2 & 1 & 0\\ 0 & 0 & 1 & 1\\ 0 & 0 & 0 & 0 \end{bmatrix}.

The equations are

x1+2x2+x3=0, x_1+2x_2+x_3=0, x3+x4=0. x_3+x_4=0.

The pivot variables are x1x_1 and x3x_3. The free variables are x2x_2 and x4x_4.

Let

x2=s,x4=t. x_2=s, \qquad x_4=t.

Then

x3=−t. x_3=-t.

From the first equation,

x1+2s−t=0, x_1+2s-t=0,

so

x1=−2s+t. x_1=-2s+t.

Thus

x=[−2s+ts−tt]=s[−2100]+t[10−11]. x= \begin{bmatrix} -2s+t\\ s\\ -t\\ t \end{bmatrix} = s \begin{bmatrix} -2\\ 1\\ 0\\ 0 \end{bmatrix} + t \begin{bmatrix} 1\\ 0\\ -1\\ 1 \end{bmatrix}.

Therefore a basis for the null space is

([−2100],[10−11]). \left( \begin{bmatrix} -2\\ 1\\ 0\\ 0 \end{bmatrix}, \begin{bmatrix} 1\\ 0\\ -1\\ 1 \end{bmatrix} \right).

So

dim⁡Null⁡(A)=2. \dim \operatorname{Null}(A)=2.

26.9 Nullity

The nullity of a matrix is the dimension of its null space:

nullity⁡(A)=dim⁡Null⁡(A). \operatorname{nullity}(A) = \dim \operatorname{Null}(A).

In the previous example,

nullity⁡(A)=2. \operatorname{nullity}(A)=2.

Nullity counts the number of independent directions that AA sends to zero.

It also equals the number of free variables in the homogeneous system Ax=0Ax=0.

26.10 Rank-Nullity

If AA is an m×nm \times n matrix, then

rank⁡(A)+nullity⁡(A)=n. \operatorname{rank}(A)+\operatorname{nullity}(A)=n.

The number nn is the dimension of the domain FnF^n.

Rank counts the number of pivot variables. Nullity counts the number of free variables. Since each column corresponds to one variable,

pivot variables+free variables=n. \text{pivot variables}+\text{free variables}=n.

This theorem is one of the central counting principles of linear algebra.

26.11 Null Space and Linear Independence

The columns of AA are linearly independent exactly when

Null⁡(A)={0}. \operatorname{Null}(A)=\{0\}.

To see this, write

A=[∣∣∣a1a2⋯an∣∣∣]. A= \begin{bmatrix} |&|&&|\\ a_1&a_2&\cdots&a_n\\ |&|&&| \end{bmatrix}.

Then

Ax=x1a1+x2a2+⋯+xnan. Ax=x_1a_1+x_2a_2+\cdots+x_na_n.

The equation

Ax=0 Ax=0

is exactly the equation

x1a1+x2a2+⋯+xnan=0. x_1a_1+x_2a_2+\cdots+x_na_n=0.

Thus nonzero solutions correspond to nontrivial linear relations among the columns.

Therefore:

Null spaceColumns of AA
Null⁡(A)={0}\operatorname{Null}(A)=\{0\}Linearly independent
Null⁡(A)≠{0}\operatorname{Null}(A)\neq\{0\}Linearly dependent

26.12 Null Space and Injectivity

The linear map

T(x)=Ax T(x)=Ax

is injective exactly when

Null⁡(A)={0}. \operatorname{Null}(A)=\{0\}.

If TT is injective, then only 00 can map to 00.

Conversely, suppose

Null⁡(A)={0}. \operatorname{Null}(A)=\{0\}.

If

Au=Av, Au=Av,

then

A(u−v)=0. A(u-v)=0.

Thus

u−v∈Null⁡(A). u-v \in \operatorname{Null}(A).

So

u−v=0, u-v=0,

and hence

u=v. u=v.

Therefore TT is injective.

The null space measures the failure of injectivity.

26.13 Null Space and Information Loss

A matrix transformation may lose information.

If

Ax=0 Ax=0

for some nonzero xx, then the transformation cannot distinguish between vv and v+xv+x, because

A(v+x)=Av. A(v+x)=Av.

The null space contains exactly the directions along which information is lost.

If the null space is large, many different inputs have the same output. If the null space is trivial, different inputs have different outputs.

26.14 Null Space and Solution Sets

Consider a nonhomogeneous system

Ax=b. Ax=b.

If xpx_p is one particular solution, then every solution has the form

x=xp+z, x=x_p+z,

where

z∈Null⁡(A). z\in \operatorname{Null}(A).

Indeed, if

Axp=b Ax_p=b

and

Az=0, Az=0,

then

A(xp+z)=Axp+Az=b. A(x_p+z)=Ax_p+Az=b.

Conversely, if xx is any solution, then

A(x−xp)=Ax−Axp=b−b=0. A(x-x_p)=Ax-Ax_p=b-b=0.

Thus

x−xp∈Null⁡(A). x-x_p\in \operatorname{Null}(A).

Therefore the complete solution set is

xp+Null⁡(A). x_p+\operatorname{Null}(A).

The null space gives the homogeneous part of every linear system.

26.15 Affine Solution Spaces

When b≠0b\neq 0, the solution set of

Ax=b Ax=b

is generally not a subspace. It may fail to contain the zero vector.

Instead, if it is nonempty, it is a translate of the null space:

xp+Null⁡(A). x_p+\operatorname{Null}(A).

Such a set is called an affine subspace.

For example, a line not passing through the origin is not a vector subspace, but it can be a translate of a one-dimensional null space.

Thus homogeneous systems give subspaces. Nonhomogeneous systems give affine spaces when they are consistent.

26.16 Null Space of Special Matrices

Some matrices have immediate null spaces.

MatrixNull space
Zero matrix 0m×n0_{m\times n}All of FnF^n
Identity matrix InI_n{0}\{0\}
Invertible square matrix AA{0}\{0\}
Projection matrixDirections projected away
Matrix with repeated columnsContains nonzero relations among those columns

For the zero matrix,

0x=0 0x=0

for every xx, so every vector lies in the null space.

For the identity matrix,

Ix=x, I x=x,

so

Ix=0 Ix=0

only when

x=0. x=0.

26.17 Left Null Space

For an m×nm\times n matrix AA, the left null space is

Null⁡(AT). \operatorname{Null}(A^T).

Since ATA^T is an n×mn\times m matrix,

Null⁡(AT)⊆Fm. \operatorname{Null}(A^T)\subseteq F^m.

It consists of all vectors y∈Fmy\in F^m such that

ATy=0. A^Ty=0.

Equivalently,

yTA=0. y^TA=0.

The left null space contains vectors orthogonal to every column of AA. Later, this will be expressed as

Null⁡(AT)=Col⁡(A)⊥ \operatorname{Null}(A^T)=\operatorname{Col}(A)^\perp

when an inner product is available.

26.18 Null Space and Orthogonality

For real matrices with the standard dot product,

Null⁡(A) \operatorname{Null}(A)

is orthogonal to the row space of AA.

Indeed, Ax=0Ax=0 means each row of AA has dot product zero with xx. If the rows are

r1,…,rm, r_1,\ldots,r_m,

then

ri⋅x=0 r_i\cdot x=0

for every ii.

Since xx is orthogonal to every row, it is orthogonal to every linear combination of the rows.

Thus

Null⁡(A)=Row⁡(A)⊥. \operatorname{Null}(A)=\operatorname{Row}(A)^\perp.

This relationship belongs to the four fundamental subspaces.

26.19 Common Mistakes

A common mistake is to place the null space in the wrong ambient space.

If AA is m×nm\times n, then

Ax Ax

is defined for

x∈Fn. x\in F^n.

Therefore

Null⁡(A)⊆Fn. \operatorname{Null}(A)\subseteq F^n.

The column space, by contrast, lies in FmF^m.

Another common mistake is to confuse the null space with the set of zero columns. The null space is a set of input vectors xx, not a set of columns of AA.

A third mistake is to forget that the null space is always a subspace, while the solution set of Ax=bAx=b with b≠0b\neq0 is generally only an affine translate.

26.20 Summary

The null space of a matrix is the set of all input vectors sent to zero. It is the solution space of the homogeneous system Ax=0Ax=0, and it is a subspace of the domain.

The key ideas are:

ConceptMeaning
Null space{x:Ax=0}\{x:Ax=0\}
KernelAnother name for null space of a linear map
Trivial solutionx=0x=0
Nontrivial solutionA nonzero vector in the null space
NullityDimension of the null space
Rank-nullityrank⁡(A)+nullity⁡(A)=n\operatorname{rank}(A)+\operatorname{nullity}(A)=n
Injectivity testAA is injective exactly when Null⁡(A)={0}\operatorname{Null}(A)=\{0\}
General solutionxp+Null⁡(A)x_p+\operatorname{Null}(A)
Orthogonal relationNull⁡(A)=Row⁡(A)⊥\operatorname{Null}(A)=\operatorname{Row}(A)^\perp

The null space records the directions lost by a linear transformation. It explains free variables, nonunique solutions, column dependence, and the geometry of homogeneous systems.