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Null Space Calculator

Null Space Calculator
Calculate the null space (kernel) of a matrix

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What is the Null Space of a Matrix?

The null space (or kernel) of a matrix A is the set of all vectors x for which Ax = 0. In other words, it consists of all solutions to the homogeneous system of linear equations Ax = 0. The null space is a fundamental concept in linear algebra with applications in various fields including physics, engineering, computer science, and data analysis.

Properties of the Null Space

  • The null space of an m × n matrix A is a subspace of ℝⁿ (or ℂⁿ if working with complex numbers).
  • The dimension of the null space is called the nullity of the matrix.
  • The Rank-Nullity Theorem states that rank(A) + nullity(A) = n, where n is the number of columns in A.
  • If A is invertible (i.e., has full rank), then its null space contains only the zero vector.
  • The null space remains unchanged under elementary row operations on the matrix.

How to Find the Null Space

To find the null space of a matrix A:

  1. Row reduce the matrix A to its reduced row echelon form (RREF).
  2. Identify the free variables (those that don't correspond to pivot columns).
  3. Express the basic variables in terms of the free variables.
  4. Write the general solution as a linear combination of vectors, where each vector corresponds to setting one free variable to 1 and the others to 0.

Example

Consider the matrix:

A = [1 2 3; 4 5 6; 7 8 9]

The RREF of A is:

RREF(A) = [1 0 -1; 0 1 2; 0 0 0]

The third column corresponds to a free variable z. We can express x and y in terms of z:

x = z
y = -2z

So the general solution is:

[x; y; z] = z[1; -2; 1]

The null space is spanned by the vector [1; -2; 1], and the nullity is 1.

Applications of Null Spaces

  • Differential Equations: The null space of a differential operator corresponds to the solution of the homogeneous differential equation.
  • Network Analysis: In circuit theory, the null space of the incidence matrix represents the fundamental cycles in the network.
  • Computer Graphics: Null spaces are used in transformations and projections.
  • Data Analysis: In Principal Component Analysis (PCA), the null space can represent the dimensions with zero variance.
  • Control Theory: The null space is important in analyzing controllability and observability of systems.

Relationship to Other Concepts

The null space is closely related to other important concepts in linear algebra:

  • Column Space: The column space of A is orthogonal to the null space of A^T (the transpose of A).
  • Row Space: The row space of A is orthogonal to the null space of A.
  • Eigenspaces: The null space of (A - λI) is the eigenspace corresponding to the eigenvalue λ.
  • Singular Value Decomposition: The null space corresponds to the right singular vectors with zero singular values.

Using the Null Space Calculator

Our null space calculator helps you find the null space of any matrix:

  1. Enter the dimensions of your matrix
  2. Input the matrix elements
  3. Click "Calculate Null Space"
  4. View the basis vectors of the null space and the step-by-step solution