Null Space Calculator

Kernel of A: Ax = 0

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About Null Space Calculator

A null space calculator that finds all solutions to Ax = 0. Row reduces to RREF, identifies free variables, and constructs basis vectors for the kernel. Shows nullity (dimension). Select from preset matrices. All calculations are client-side. Essential for understanding solution spaces.

Null Space Calculator Features

  • Basis vectors
  • Nullity
  • RREF
  • Free vars
  • Presets
Null space (kernel): N(A) = {x : Ax = 0}. Always a subspace. Dimension = nullity = n − rank. Found by row reducing [A] to RREF, then expressing free variables in terms of each other. Each free variable gives one basis vector.

How to Use

Enter a matrix:

  • A: Any m×n matrix
  • Output: Null space basis
  • Nullity: Dimension

Significance

Null space = 0 iff A is full column rank. Non-trivial null space means columns are dependent. For square matrices: null space ≠ {0} ⟺ A is singular ⟺ det(A) = 0.

Applications

  • General solution: x = x_p + x_null
  • Linear dependence relations
  • Eigenspace (null space of A−λI)

Step-by-Step Instructions

  1. 1Enter a matrix.
  2. 2View RREF.
  3. 3Identify free variables.
  4. 4Get null space basis.
  5. 5Check nullity.

Null Space Calculator — Frequently Asked Questions

How is null space related to eigenspaces?+

The eigenspace for eigenvalue λ is the null space of (A−λI). Finding eigenvectors = finding null space of a shifted matrix. If λ=0 is an eigenvalue, its eigenspace IS the null space of A.

What if the null space is trivial?+

N(A) = {0} means only the zero vector solves Ax=0. This happens iff rank(A) = n (full column rank). For square matrices, this means A is invertible. For rectangular m×n with m>n, this is the 'overdetermined, independent' case.

How does null space relate to the general solution?+

If x_p is any particular solution to Ax=b, then ALL solutions are x = x_p + x_h where x_h ∈ N(A). The null space is the 'freedom' in the solution. More null space dimensions = more freedom.

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