Калькулятор Нулевого Пространства

Найдите нулевое пространство (ядро) любой матрицы, решая Ax = 0 методом исключения Гаусса. Получите базисные векторы, размерность ядра, пошаговое приведение к RREF и проверку теоремы о ранге и дефекте с точными дробными вычислениями.

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О Калькулятор Нулевого Пространства

The Null Space Calculator finds the null space (kernel) of any matrix by solving the homogeneous system Ax = 0. Enter a matrix of any size up to 8×8 and get the complete null space basis with exact fractional arithmetic, step-by-step Gaussian elimination to RREF, column classification (pivot vs. free), and rank-nullity theorem verification.

What Is the Null Space of a Matrix?

The null space (also called the kernel) of an $m \times n$ matrix $A$ is the set of all vectors $\mathbf{x}$ in $\mathbb{R}^n$ that satisfy:

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

Written as a set: $\text{Null}(A) = \{ \mathbf{x} \in \mathbb{R}^n : A\mathbf{x} = \mathbf{0} \}$. The null space is always a subspace of $\mathbb{R}^n$, meaning it contains the zero vector and is closed under addition and scalar multiplication.

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Subspace

The null space is always a subspace of R^n. It contains the zero vector and is closed under addition and scalar multiplication.

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Dimension = Nullity

The dimension of the null space equals the nullity — the number of free variables after row reduction.

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Free Variables

Each free variable in the RREF contributes one basis vector to the null space.

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Rank-Nullity

rank(A) + nullity(A) = n, where n is the number of columns. This always holds.

How to Find the Null Space

Step 1. Set the number of rows (m) and columns (n) for your matrix using the +/− controls, or click a quick example to load a preset matrix.

Step 2. Enter your matrix values into the grid. You can type integers, decimals, or fractions like 1/3 or -5/2. Use Tab, Enter, or arrow keys to navigate between cells.

Step 3. Click Find Null Space. The calculator performs Gaussian elimination to convert your matrix to reduced row echelon form (RREF).

Step 4. Identify pivot columns and free columns. Each free column corresponds to a free variable that can take any value.

Step 5. For each free variable, set it to 1 and all other free variables to 0, then solve for the pivot variables. The resulting vectors form a basis for the null space.

Null Space vs. Column Space

Property	Null Space	Column Space
Definition	All x such that Ax = 0	All b such that Ax = b has a solution
Lives in	$\mathbb{R}^n$ (domain)	$\mathbb{R}^m$ (codomain)
Dimension	nullity = n − rank	rank
Found from	Free columns of RREF	Pivot columns of A

The Rank-Nullity Theorem

For any $m \times n$ matrix $A$:

$$\text{rank}(A) + \text{nullity}(A) = n$$

The rank is the number of pivot columns in the RREF, and the nullity is the number of free columns. Together they account for every column. This theorem is also known as the dimension theorem for linear maps.

Special Cases

Scenario	Null Space	What It Means
Full column rank (rank = n)	Only {0}	Columns are linearly independent; Ax = 0 has only the trivial solution
More columns than rows (n > m)	Always nontrivial	There are at least n − m free variables, so infinitely many solutions exist
Square singular matrix	Nontrivial	The matrix has a zero determinant and dependent rows/columns
Zero matrix	All of $\mathbb{R}^n$	Every vector is in the null space; the basis is the standard basis

Applications of the Null Space

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Linear Systems

The general solution to Ax = b is x = x_p + Null(A), where x_p is one particular solution.

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Linear Independence

Columns of A are linearly independent if and only if Null(A) = {0}.

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Differential Equations

The solution space of a homogeneous linear ODE system is the null space of the coefficient matrix.

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Error-Correcting Codes

In coding theory, valid codewords form the null space of the parity-check matrix H.

Frequently Asked Questions

What is the null space of a matrix?

The null space (or kernel) of a matrix A is the set of all vectors x such that Ax = 0. It is a subspace of R^n where n is the number of columns. The null space always contains the zero vector and may also contain infinitely many nonzero vectors if the matrix has free variables.

How do you find the null space?

Reduce the matrix A to reduced row echelon form (RREF) using Gaussian elimination. Identify pivot columns and free columns. For each free variable, set it to 1 and all other free variables to 0, then solve for the pivot variables. The resulting vectors form a basis for the null space.

What is the rank-nullity theorem?

The rank-nullity theorem states that for an m by n matrix A, rank(A) + nullity(A) = n, where n is the number of columns. The rank is the number of pivot columns and the nullity is the dimension of the null space (number of free variables).

What does it mean if the null space is trivial?

A trivial null space means the only solution to Ax = 0 is the zero vector x = 0. This happens when every column is a pivot column (full column rank). It means the columns of A are linearly independent.

Can non-square matrices have a null space?

Yes. Any matrix has a null space. For an m by n matrix with m less than n, the null space is guaranteed to be nontrivial (dimension at least n - m) because there are more unknowns than equations, so free variables always exist.

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by miniwebtool team. Updated: 2026-04-10

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