Homogeneous Systems: Ax = 0 & Null Space

Definition

A homogeneous linear system is one where every equation has zero on the right-hand side:

A\mathbf{x} = \mathbf{0}

The augmented matrix is

[A \mid \mathbf{0}]

. Since the last column is all zeros, row operations on the augmented matrix never produce a contradictory row

[0 \; \cdots \; 0 \mid d]

with

d \neq 0

. A homogeneous system is always consistent.

The vector

\mathbf{x} = \mathbf{0}

satisfies every equation — this is the trivial solution. It always exists. The central question for a homogeneous system is never "does a solution exist?" but "does a nontrivial solution exist?"

Homogeneous System Notation

Notation

Homogeneous System Notation

One boldface character carries the whole definition: the right-hand side is a vector of zeros, not the number zero. From that follow a solution that is guaranteed rather than sought, and a vocabulary that names the two cases a homogeneous system can be in. All of it is catalogued among the linear algebra symbols.

The augmented bar

[A \mid \mathbf{0}]

and the row-operation conventions come from Gaussian elimination notation;

\operatorname{Nul}(A)

from fundamental-space notation;

\operatorname{rank}

and the free-variable count from rank and echelon form notation;

\det

from determinant notation.

A\mathbf{x} = \mathbf{0}

A x equals the zero vector

The boldface is the definition, not styling:

\mathbf{0}

is a column of

m

zeros, while a plain

0

is a single number. Writing

A\mathbf{x} = 0

equates a vector with a scalar — the shapes do not match, and the equation only typechecks once the right side is bold.

Also written

\vec{0}

where the site's arrow convention is in force, and

\mathbf{0}_m

when the length needs stating; handwriting has no bold, so an underline or an arrow does the work at the board — which is exactly where the scalar/vector slip happens.

Do not confuseThe zero matrix

O

. Same idea, different shape:

O

is a rectangular array of zeros,

\mathbf{0}

a single column, and matrix operation notation reserves the two letters separately for that reason.

\mathbf{x} = \mathbf{0}

the trivial solution

A named solution rather than a described one:

\mathbf{x} = \mathbf{0}

satisfies

A\mathbf{x} = \mathbf{0}

for every

A

, so it is guaranteed, never discovered. As Definition above puts it, the question for a homogeneous system is never whether a solution exists but whether another one does.

Cases"Nontrivial" is the entire vocabulary for the interesting case, and it is a claim about existence of a second solution, not about complexity — a nontrivial solution can be as simple as

(1, 0, 0)

. Its presence is equivalent to a free column, to

\operatorname{rank}(A) < n

, and for square

A

to

\det(A) = 0

.

Do not confuse"Trivial" as a comment on difficulty. Here it is a technical label for one specific vector, not an editorial judgement — and a system whose only solution is trivial is the more constrained, often harder case.

\{\mathbf{x} : A\mathbf{x} = \mathbf{0}\} = \operatorname{Nul}(A)

the solution set is the null space of A

An equality between two descriptions of one object: set-builder on the left, a named fundamental space on the right. Only homogeneous systems earn this — the notation records that the solutions form a subspace rather than a scattered set.

CasesThe Superposition Principle below is that subspace closure spelled out: if

\mathbf{u}

and

\mathbf{v}

solve the system then so does

c\mathbf{u} + d\mathbf{v}

, which is why the general solution can be written as a span rather than a list.

Do not confuseExpecting the same of

A\mathbf{x} = \mathbf{b}

. A non-homogeneous solution set is a translate of

\operatorname{Nul}(A)

, not a subspace — it misses

\mathbf{0}

unless

\mathbf{b} = \mathbf{0}

, which is precisely the structural difference Homogeneous vs. Non-Homogeneous below turns on.

All linear algebra symbols →↑ Linear Systems

When Do Nontrivial Solutions Exist?

Nontrivial solutions to

A\mathbf{x} = \mathbf{0}

exist if and only if the rank of

A

is less than

n

, the number of unknowns. When

\text{rank}(A) < n

, at least one free variable appears in the echelon form, and that free variable parametrizes a family of nonzero solutions.

For a square

n \times n

matrix, nontrivial solutions exist if and only if

\det(A) = 0

. A nonzero determinant means full rank, which means no free variables, which means only the trivial solution.

One case is automatic: if the system has more unknowns than equations (

n > m

), nontrivial solutions always exist:

Underdetermined Homogeneous Has Nontrivial

m < n \Rightarrow A\mathbf{x} = \mathbf{0} \text{ has nontrivial solutions}

Underdetermined homogeneous has nontrivial →

The rank of an

m \times n

matrix cannot exceed

m

, and when

m < n

, the rank is strictly less than

n

. This guarantees at least

n - m

free variables, producing an infinite family of nontrivial solutions. Fewer equations than unknowns always leaves room.

Setting	Condition for nontrivial solutions to A x = 0	Equivalent statements
General m × n matrix	rank(A) < n	nullity(A) > 0; at least one free variable in the echelon form of A
Square n × n matrix	det(A) = 0	A is singular; A is not invertible; columns are linearly dependent
Wide matrix (n > m)	always — nontrivial solutions guaranteed	rank(A) ≤ m < n, so n − rank(A) ≥ n − m ≥ 1 free variables remain

A free column, and with it a family of solutions

The elimination has left a column without a pivot, so one unknown can be chosen freely and the others follow. For a homogeneous system that guarantees solutions beyond the zero one, which is why more unknowns than equations always produces them — there are not enough pivots to go round. Vary the shape and watch the free columns appear on the linear system solutions visualizer.

Counting pivots against unknowns therefore answers the question before any solution is written down.

The Solution Set Is the Null Space

The set of all solutions to

A\mathbf{x} = \mathbf{0}

is the null space of

A

:

\text{Null}(A) = \{\mathbf{x} \in \mathbb{R}^n : A\mathbf{x} = \mathbf{0}\}

The null space is a subspace of

\mathbb{R}^n

. It contains

\mathbf{0}

, and it is closed under addition and scalar multiplication: if

A\mathbf{u} = \mathbf{0}

and

A\mathbf{v} = \mathbf{0}

, then

A(\mathbf{u} + \mathbf{v}) = \mathbf{0}

and

A(c\mathbf{u}) = \mathbf{0}

.

The dimension of the null space is the nullity:

Homogeneous Solution Space Dimension

\dim(\text{Null}(A)) = n - \text{rank}(A)

Homogeneous solution space dimension →

When the nullity is

0

, the null space is

\{\mathbf{0}\}

and only the trivial solution exists. When the nullity is

k > 0

, the null space is a

k

-dimensional subspace, and the solution set contains infinitely many vectors forming a

k

-dimensional flat through the origin.

Special solutions, one per free column

Each free variable is set to one in turn while the others are held at zero, and the pivot variables are solved to match. The vectors produced are independent and they span every solution of the system, which is what it means to call the solution set the null space rather than merely to say it resembles one. Generate them for your own matrix on the four fundamental subspaces visualizer.

Because the set is closed under addition and scaling, it is a subspace, and that is what makes a basis for it worth having.

Finding the Null Space

The algorithm is a direct application of Gaussian elimination. Row reduce

A

to echelon form (reducing just

A

— the zero augmented column adds nothing). Identify the pivot variables and the free variables. For each free variable, set it to

1

with all other free variables at

0

, and solve for the pivot variables by back substitution. Each setting produces one basis vector for the null space.

Worked Example

A = \begin{pmatrix} 1 & 2 & -1 & 0 & 3 \\ 2 & 4 & 0 & 2 & 8 \\ -1 & -2 & 2 & 1 & 0 \end{pmatrix}

Row reduce:

\xrightarrow{R_2 - 2R_1,\; R_3 + R_1} \begin{pmatrix} 1 & 2 & -1 & 0 & 3 \\ 0 & 0 & 2 & 2 & 2 \\ 0 & 0 & 1 & 1 & 3 \end{pmatrix} \xrightarrow{R_3 - \frac{1}{2}R_2} \begin{pmatrix} 1 & 2 & -1 & 0 & 3 \\ 0 & 0 & 2 & 2 & 2 \\ 0 & 0 & 0 & 0 & 2 \end{pmatrix}

Pivots in columns

1

,

3

,

5

. Free variables:

x_2 = s

,

x_4 = t

. Row

3

:

2x_5 = 0 \Rightarrow x_5 = 0

. Row

2

:

2x_3 + 2t = 0 \Rightarrow x_3 = -t

. Row

1

:

x_1 + 2s + t + 0 = 0 \Rightarrow x_1 = -2s - t

.

Setting

s = 1, t = 0

:

\mathbf{v}_1 = (-2, 1, 0, 0, 0)

. Setting

s = 0, t = 1

:

\mathbf{v}_2 = (-1, 0, -1, 1, 0)

.

The null space is

\text{Span}\{\mathbf{v}_1, \mathbf{v}_2\}

, a two-dimensional subspace of

\mathbb{R}^5

.

Parametric Vector Form

The general solution to

A\mathbf{x} = \mathbf{0}

is a linear combination of the null-space basis vectors:

\mathbf{x} = t_1\mathbf{v}_1 + t_2\mathbf{v}_2 + \cdots + t_k\mathbf{v}_k

where

\mathbf{v}_1, \dots, \mathbf{v}_k

are the basis vectors found by the algorithm above and

t_1, \dots, t_k

are free parameters ranging over all real numbers. The number of parameters

k = n - \text{rank}(A)

is the nullity.

When

k = 0

, the only solution is

\mathbf{x} = \mathbf{0}

. When

k = 1

, the solutions form a line through the origin in

\mathbb{R}^n

. When

k = 2

, a plane through the origin. In general, the solution set is a

k

-dimensional subspace passing through the origin.

There is no particular solution

\mathbf{x}_p

to add because the right-hand side is

\mathbf{0}

— the zero vector is itself the particular solution. The entire solution set is the null space, unshifted.

The Superposition Principle

If

\mathbf{x}_1

and

\mathbf{x}_2

are solutions to

A\mathbf{x} = \mathbf{0}

, then any linear combination

c_1\mathbf{x}_1 + c_2\mathbf{x}_2

is also a solution:

A(c_1\mathbf{x}_1 + c_2\mathbf{x}_2) = c_1 A\mathbf{x}_1 + c_2 A\mathbf{x}_2 = c_1\mathbf{0} + c_2\mathbf{0} = \mathbf{0}

This is precisely the statement that the solution set is a subspace — it is closed under addition and scalar multiplication. The superposition principle is the reason the general solution is a linear combination of basis vectors, and it is the reason the null space has the clean structure of a vector space rather than an arbitrary collection of points.

Superposition holds only for homogeneous systems. For a non-homogeneous system

A\mathbf{x} = \mathbf{b}

with

\mathbf{b} \neq \mathbf{0}

, the sum of two solutions is generally not a solution:

A(\mathbf{x}_1 + \mathbf{x}_2) = \mathbf{b} + \mathbf{b} = 2\mathbf{b} \neq \mathbf{b}

.

Homogeneous vs. Non-Homogeneous

The homogeneous system

A\mathbf{x} = \mathbf{0}

and the non-homogeneous system

A\mathbf{x} = \mathbf{b}

are deeply connected. If

\mathbf{x}_p

is any particular solution to

A\mathbf{x} = \mathbf{b}

, then every solution has the form

\mathbf{x} = \mathbf{x}_p + \mathbf{x}_h

where

\mathbf{x}_h \in \text{Null}(A)

is a solution to the homogeneous system. The particular solution accounts for

\mathbf{b}

; the null-space component accounts for the freedom.

This decomposition has two immediate consequences. If the null space is trivial (

\text{nullity} = 0

), the non-homogeneous system has at most one solution — either

\mathbf{x}_p

alone or nothing. If the null space is nontrivial (

\text{nullity} > 0

), then either

A\mathbf{x} = \mathbf{b}

has no solution or it has infinitely many — there is no middle ground.

The solution set of

A\mathbf{x} = \mathbf{b}

is therefore a translated copy of the null space: the null space shifted by

\mathbf{x}_p

. The homogeneous system determines the shape and dimension of the solution set; the particular solution determines its position.

Aspect	Homogeneous: A x = 0	Non-homogeneous: A x = b, b ≠ 0
Always consistent?	yes — the trivial solution x = 0 always works	no — depends on whether b ∈ Col(A)
Solution set	Null(A) — a subspace passing through the origin	x_p + Null(A) — an affine flat shifted away from the origin
Superposition	closed: any linear combination of solutions is a solution	not closed: A(x₁ + x₂) = 2b ≠ b
When uniquely solvable	always uniquely "solvable" — only the trivial solution when rank(A) = n	when rank(A) = rank([A \| b]) = n
Number of solutions	just the trivial one, or infinitely many (never zero, never finite > 1)	zero, one, or infinitely many

Homogeneous Systems and Linear Independence

Testing whether vectors

\mathbf{v}_1, \dots, \mathbf{v}_k

are linearly independent is equivalent to checking whether the homogeneous system

A\mathbf{c} = \mathbf{0}

has only the trivial solution, where

A = [\mathbf{v}_1 \; \mathbf{v}_2 \; \cdots \; \mathbf{v}_k]

.

The equation

c_1\mathbf{v}_1 + c_2\mathbf{v}_2 + \cdots + c_k\mathbf{v}_k = \mathbf{0}

is literally the system

A\mathbf{c} = \mathbf{0}

. If the null space of

A

is trivial, the only solution is

\mathbf{c} = \mathbf{0}

and the vectors are independent. If the null space is nontrivial, some nonzero

\mathbf{c}

satisfies the equation, providing an explicit dependence relation — the entries of

\mathbf{c}

are the coefficients that express one vector as a combination of the others.

This is the computational link between homogeneous systems and independence. Row reducing

A

and checking for free variables is the standard algorithm for deciding independence, and the null-space basis vectors encode the dependence relations when they exist.

The Eigenvalue Connection

The eigenvalue equation

A\mathbf{x} = \lambda\mathbf{x}

can be rewritten as

(A - \lambda I)\mathbf{x} = \mathbf{0}

This is a homogeneous system with coefficient matrix

A - \lambda I

. Eigenvectors are precisely the nontrivial solutions. They exist when and only when

A - \lambda I

is singular — that is, when

\det(A - \lambda I) = 0

.

The values of

\lambda

satisfying this determinant condition are the eigenvalues. For each eigenvalue

\lambda

, the set of all solutions to

(A - \lambda I)\mathbf{x} = \mathbf{0}

is the eigenspace — the null space of

A - \lambda I

. The dimension of this eigenspace is the nullity of

A - \lambda I

, which equals

n - \text{rank}(A - \lambda I)

.

This rewriting connects homogeneous systems directly to spectral theory. Every eigenvalue problem is, at its core, a question about when a particular homogeneous system has nontrivial solutions. The machinery of row reduction, null spaces, and rank that governs homogeneous systems is the same machinery that computes eigenvectors and eigenspaces.

Summary: Where Homogeneous Systems Appear

The homogeneous system A·x = 0 shows up far beyond its own statement: every test of linear independence, every eigenvector calculation, every analysis of a non-homogeneous solution set, and every kernel computation reduces to solving one. The table below collects each construction in which homogeneous systems play a role, alongside the specific equation involved and what its solutions represent in that context.

The six settings below are not applications of homogeneous systems so much as the same system wearing different names. In each case something is being asked about a null space, and the only thing that changes is what the trivial solution is taken to mean — no redundancy among vectors, no eigenvector for that scalar, no collapse under the transformation.

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Linear systems · homogeneous

Where $A\mathbf{x} = \mathbf{0}$ turns up

Six questions from six different chapters, all of which reduce to solving one homogeneous system. What differs is only what the null space is being called and what its being trivial is taken to mean.

6settings

The system itself2

1

Solution set§ 3

always consistent —

\mathbf{x} = \mathbf{0}

works

A\mathbf{x} = \mathbf{0}

The solution set is

\operatorname{Null}(A)

, a subspace rather than merely a set: closed under addition and scaling, which is the superposition principle. One basis vector per free variable, so the dimension is

n - \operatorname{rank}(A)

.

2

Non-homogeneous solution set§ 7

when

A\mathbf{x} = \mathbf{b}

is consistent

\mathbf{x} = \mathbf{x}_p + \operatorname{Null}(A)

The homogeneous solution supplies the entire shape and size of the general solution; the particular solution only translates it off the origin. This is why the two systems are never solved independently — the same reduction gives both.

The same equation, asked differently4

3

Linear independence§ 8

independent

\iff

only

\mathbf{c} = \mathbf{0}

A\mathbf{c} = \mathbf{0}

, columns

= \mathbf{v}_i

Independence is the statement that this system has only the trivial solution. When it does not, the nontrivial

\mathbf{c}

is not merely evidence of dependence — its entries are the coefficients of an explicit dependence relation among the vectors.

4

Column dependence relations§ 8

one relation per null space basis vector

nontrivial

\mathbf{c}

with

A\mathbf{c} = \mathbf{0}

Read the same solution the other way round: each entry of

\mathbf{c}

says how much of the corresponding column enters the relation, so one column is written as a combination of the others. Which columns are redundant is exactly which are non-pivot.

5

Eigenvectors§ 9

\lambda

already known

(A - \lambda I)\mathbf{x} = \mathbf{0}

The eigenspace for

\lambda

is

\operatorname{Null}(A - \lambda I)

, so finding eigenvectors is solving a homogeneous system. A nontrivial solution must exist, which is precisely why

\det(A - \lambda I) = 0

is the condition defining

\lambda

in the first place.

6

Kernel of a transformation§ 3

\ker(T) = \operatorname{Null}(A)

for the matrix of

T

T(\mathbf{x}) = \mathbf{0}

The abstract statement of the same thing.

T

is injective exactly when the kernel is trivial — so the size of the null space measures how far a transformation is from being one-to-one.

One computation serves all six: reduce, identify the free variables, read off a basis. That is why the homogeneous case is worth treating first — the non-homogeneous system, the independence test, the eigenvector calculation and the kernel are the same reduction asked about differently.

Where Ax = 0 turns up·/linear-algebra/linear-systems/homogeneousLearn Math Class

Two of these are worth reading together, because they are the same solution used twice. Solving

A\mathbf{c} = \mathbf{0}

answers whether the columns are independent, and when the answer is no, the vector

\mathbf{c}

is not merely a witness to dependence — its entries are the coefficients of the relation. The test and the explanation come from one computation, which is why nothing further is needed once the null space basis is in hand.

The eigenvector row is where the ordering matters.

(A - \lambda I)\mathbf{x} = \mathbf{0}

is only worth solving once

\lambda

is known, and

\lambda

is chosen precisely so that this system has a nontrivial solution — which is what

\det(A - \lambda I) = 0

asserts. The characteristic equation exists to make the homogeneous system interesting; the eigenvector calculation that follows is ordinary null space work.

Homogeneous Systems FAQ

Does "trivial solution" mean the easy one?

+

No, it is a technical label for one specific vector, the all-zeros solution that every homogeneous system has. Nothing about difficulty is implied, and a system whose only solution is the trivial one is the more constrained case. Similarly, nontrivial claims that a second solution exists, not that it is complicated:

(1, 0, 0)

qualifies.Read more →

Is the solution set of $A\mathbf{x} = \mathbf{b}$ a subspace?

+

Only when

\mathbf{b} = \mathbf{0}

. A homogeneous solution set is the null space and is closed under addition and scaling, which is why the general solution can be written as a span. A non-homogeneous set is a translate of that null space: it has the same shape but misses the origin, so it fails the first subspace test.Read more →

What is the difference between $\mathbf{0}$ and $O$ ?

+

Shape. Bold

\mathbf{0}

is the zero vector, a single column, while

O

is the zero matrix, a rectangular array. The two letters are reserved separately precisely so an equation can hold both without ambiguity, and the bold face is what distinguishes the vector from the plain scalar zero as well.Read more →

Homogeneous Systems of Equations

Systems with Zero Right-Hand Side

Definition

Homogeneous System Notation

Homogeneous System Notation

When Do Nontrivial Solutions Exist?

The Solution Set Is the Null Space

Finding the Null Space

Worked Example

Parametric Vector Form

The Superposition Principle

Homogeneous vs. Non-Homogeneous

Homogeneous Systems and Linear Independence

The Eigenvalue Connection

Summary: Where Homogeneous Systems Appear

Where $A\mathbf{x} = \mathbf{0}$ turns up

Homogeneous Systems FAQ

Does "trivial solution" mean the easy one?

Is the solution set of $A\mathbf{x} = \mathbf{b}$ a subspace?

What is the difference between $\mathbf{0}$ and $O$ ?

Homogeneous Systems of Equations

Systems with Zero Right-Hand Side

Definition

Homogeneous System Notation

Homogeneous System Notation

When Do Nontrivial Solutions Exist?

The Solution Set Is the Null Space

Finding the Null Space

Worked Example

Parametric Vector Form

The Superposition Principle

Homogeneous vs. Non-Homogeneous

Homogeneous Systems and Linear Independence

The Eigenvalue Connection

Summary: Where Homogeneous Systems Appear

Where Ax=0A\mathbf{x} = \mathbf{0}Ax=0 turns up

Homogeneous Systems FAQ

Does "trivial solution" mean the easy one?

Is the solution set of Ax=bA\mathbf{x} = \mathbf{b}Ax=b a subspace?

What is the difference between 0\mathbf{0}0 and OOO?

Where $A\mathbf{x} = \mathbf{0}$ turns up

Is the solution set of $A\mathbf{x} = \mathbf{b}$ a subspace?

What is the difference between $\mathbf{0}$ and $O$ ?