Find Null Space Calculator

What is the Null Space?

The null space of a matrix A, also known as its kernel, is a fundamental concept in linear algebra. It consists of all vectors x that, when multiplied by the matrix A, result in the zero vector (0). Mathematically, the null space N(A) is defined as:

N(A) = {x | Ax = 0}

The null space is a vector subspace of the domain of the linear transformation represented by A. It provides crucial information about the solutions to the homogeneous system of linear equations Ax = 0. If the null space only contains the zero vector, it means the columns of A are linearly independent, and the only solution to Ax = 0 is the trivial solution x = 0. If the null space contains non-zero vectors, there are infinitely many solutions to Ax = 0.

Anyone working with systems of linear equations, linear transformations, or vector spaces, such as engineers, mathematicians, physicists, and computer scientists, might use the concept of a null space. A common misconception is that the null space is related to finding zero elements in the matrix itself; it's about the vectors that are mapped to the zero vector *by* the matrix transformation.

Null Space Formula and Mathematical Explanation

There isn't a single "formula" for the null space, but rather a procedure to find a basis for it:

1. Start with the matrix A: You have an m x n matrix A.
2. Set up the homogeneous equation: We want to solve Ax = 0.
3. Row Reduction: Perform Gaussian elimination to transform matrix A into its Reduced Row Echelon Form (RREF).
4. Identify Pivot and Free Variables: In the RREF, columns with leading 1s (pivots) correspond to pivot variables. Columns without leading 1s correspond to free variables.
5. Express Pivot Variables: Write the equations from the RREF, expressing each pivot variable in terms of the free variables.
6. Write the General Solution: Write the solution vector x with each component expressed in terms of the free variables.
7. Parametric Vector Form: Decompose the general solution into a linear combination of vectors, where each vector is multiplied by a free variable. These vectors form a basis for the null space of A.

The number of free variables is equal to the dimension of the null space, also called the nullity of A. The Rank-Nullity Theorem states that for an m x n matrix A, rank(A) + nullity(A) = n (number of columns).

Variables in Null Space Calculation

Variable	Meaning	Typical Representation
A	The m x n matrix	[aij]
x	An n x 1 column vector in the domain	[x1, x2, …, xn]^T
0	The m x 1 zero vector	[0, 0, …, 0]^T
RREF(A)	Reduced Row Echelon Form of A	Matrix with leading 1s and zeros
Pivot Variables	Variables corresponding to columns with pivots in RREF(A)	xi, xj,…
Free Variables	Variables corresponding to columns without pivots in RREF(A)	xk, xl,…
Basis Vectors	Vectors that span the null space	v1, v2,…
Nullity(A)	Dimension of the null space (number of free variables)	Integer ≥ 0

Practical Examples (Real-World Use Cases)

Example 1: A 2×3 Matrix

Let's find the null space of the matrix A = [[1, 2, 3], [2, 4, 6]].

1. We want to solve Ax = 0, so [[1, 2, 3], [2, 4, 6]] [x1, x2, x3]^T = [0, 0]^T.

2. Row reduce A: [[1, 2, 3], [2, 4, 6]] -> R2 = R2 – 2*R1 -> [[1, 2, 3], [0, 0, 0]]. This is the RREF.

3. Pivots: The first column has a pivot (1). Columns 2 and 3 do not.

4. Variables: x1 is a pivot variable, x2 and x3 are free variables.

5. Equation from RREF: 1*x1 + 2*x2 + 3*x3 = 0 => x1 = -2*x2 – 3*x3.

6. General solution: x = [x1, x2, x3]^T = [-2*x2 – 3*x3, x2, x3]^T

7. Parametric form: x = x2*[-2, 1, 0]^T + x3*[-3, 0, 1]^T. The basis for the null space is {[-2, 1, 0]^T, [-3, 0, 1]^T}. The nullity is 2.

Example 2: A 3×3 Matrix with Trivial Null Space

Consider A = [[1, 0, 0], [0, 1, 0], [0, 0, 1]] (the identity matrix).

1. Ax = 0 => x1=0, x2=0, x3=0.

2. A is already in RREF.

3. Pivots in columns 1, 2, 3.

4. x1, x2, x3 are pivot variables. No free variables.

5. x1=0, x2=0, x3=0.

6. General solution: x = [0, 0, 0]^T.

7. The null space is just the zero vector {0}. The basis is the empty set or {0}, and the nullity is 0. Our matrix rank calculator can help determine the rank first.

How to Use This Find Null Space Calculator

1. Select Matrix Dimensions: Choose the number of rows (m) and columns (n) for your matrix A using the dropdown menus. The calculator supports matrices up to 4×5.
2. Enter Matrix Elements: Input the values for each element of the matrix A into the generated input fields.
3. Calculate: Click the "Calculate Null Space" button.
4. View Results:
   • Primary Result: Shows the dimension of the null space (nullity).
   • RREF: Displays the Reduced Row Echelon Form of your matrix.
   • Pivot Columns & Free Variables: Lists the indices of pivot columns and free variables.
   • Basis Vectors: Shows the vectors that form a basis for the null space. If the null space is just the zero vector, it will indicate that.
   • Table & Chart: A table of the RREF and a chart visualizing pivot vs. free variables are also shown.
5. Interpret: The basis vectors span all possible solutions to Ax = 0. Any linear combination of these basis vectors is in the null space. The nullity tells you how many independent vectors form the basis. For more on vectors, see our vector calculator.
6. Reset: Click "Reset" to clear the inputs and results for a new calculation.

Key Factors That Affect Null Space Results

• Matrix Elements: The specific values within the matrix directly determine the relationships between rows and columns, and thus the RREF and null space.
• Number of Rows (m) and Columns (n): The dimensions of the matrix constrain the maximum possible rank and influence the nullity (n – rank).
• Linear Independence of Rows/Columns: If rows or columns are linearly dependent, it often leads to zero rows in the RREF and free variables, resulting in a non-trivial null space.
• Rank of the Matrix: The rank (number of pivots in RREF) is inversely related to the nullity (rank + nullity = n). A higher rank means a smaller null space dimension. You can use a rank calculator.
• Presence of Free Variables: The existence of free variables after row reduction directly leads to a null space with dimension greater than zero.
• Matrix being Invertible (for square matrices): If a square matrix is invertible, its rank is full, its RREF is the identity matrix, and its null space is trivial ({0}). Our matrix inverse calculator can check this.

Frequently Asked Questions (FAQ)

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's a subspace of the domain.

What is the nullity of a matrix?

The nullity of a matrix is the dimension of its null space. It equals the number of free variables in the solution to Ax = 0, or n – rank(A).

How do you find the basis of a null space?

By row-reducing the matrix to RREF, identifying free variables, and expressing the general solution to Ax = 0 in parametric vector form. The vectors multiplying the free parameters form the basis.

What does it mean if the null space is just the zero vector?

It means the only solution to Ax = 0 is x = 0 (the trivial solution). This occurs when the columns of A are linearly independent, and for a square matrix, it means the matrix is invertible.

Can the null space be empty?

No, the null space always contains at least the zero vector because A0 = 0.

Is the null space the same as the kernel?

Yes, for a linear transformation represented by a matrix, the null space of the matrix is the kernel of the transformation.

How is the null space related to the column space?

The null space is a subspace of the domain (Rⁿ), while the column space is a subspace of the codomain (R^m). For an m x n matrix, dim(null space) + dim(column space) = n (Rank-Nullity Theorem).

Does every matrix have a null space?

Yes, every m x n matrix has a null space, which is a subspace of Rⁿ.

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