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House of Calculators

Null Space Calculator

Find the Kernel of a Matrix

Calculate the null space (kernel) of a matrix - the set of all vectors that the matrix maps to the zero vector. Essential for solving homogeneous systems and understanding linear transformations in linear algebra.

Compute the Null Space of a Matrix

Enter the dimensions and elements of a matrix to calculate its null space.

Understanding Null Space

The null space of a matrix \( A \) consists of all vectors \( x \) that satisfy the equation:

A⋅x=0A \cdot x = 0

Finding the null space is crucial for understanding the solutions to homogeneous systems of linear equations. It helps in determining the linear dependencies among the columns of the matrix.

Calculating the Null Space

To calculate the null space of a matrix, follow these steps:

  1. Convert the matrix to Reduced Row Echelon Form (RREF). You can use our RREF Calculator for this step.
  2. Identify the free variables (columns without leading 1's in the RREF).
  3. Express the basic variables in terms of the free variables.
  4. Create basis vectors for the null space by setting each free variable to 1 (with others at 0) and solving for the basic variables.

Interpreting the Results

When you use this null space calculator:

  • If the result is empty, the null space is trivial (contains only the zero vector).
  • If one or more vectors are returned, these form a basis for the null space. Any linear combination of these vectors will also be in the null space.
  • The number of basis vectors gives the nullity of the matrix, which is the dimension of the null space.

Applications

  • Solving homogeneous systems of linear equations.
  • Understanding linear transformations and their properties.
  • Analyzing the structure of linear operators in quantum mechanics.
  • Optimizing processes in control theory and systems engineering.
  • Solving underdetermined systems in data compression and image processing.

Relation to Other Concepts

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

  • Rank: The nullity of a matrix plus its rank equals the number of columns (Rank-Nullity theorem).
  • Column Space: The null space of AT (transpose of A) is the orthogonal complement of the column space of A.
  • Eigenspaces: The null space of (A - λI) is the eigenspace of A corresponding to the eigenvalue λ.

Understanding the null space and its properties is essential for a deep comprehension of linear algebra and its applications in various fields of science and engineering.

What is Null Space?

The null space of matrix A is the set of all vectors x where Ax = 0. Also called the kernel, it's a subspace of the domain. Finding null space: row reduce A to RREF, identify free variables, express solution as linear combination of basis vectors. Dimension of null space is the nullity.

Rank-Nullity Theorem

For an m×n matrix A: rank(A) + nullity(A) = n. This fundamental theorem relates the dimension of the column space (rank) to the dimension of the null space (nullity). A full rank matrix (rank = n) has trivial null space (only zero vector).

Applications

Null space is used in: solving homogeneous systems Ax=0, finding linear dependencies in vectors, analyzing linear transformations, computer graphics and 3D modeling, differential equations solutions, and eigenvalue problems. Understanding null space is crucial for advanced linear algebra and applications.