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Null Space Calculator

Use this free null space calculator to find the null space (kernel) of a 2x2 matrix. Determine whether the null space is trivial or non-trivial, get the nullity, rank, and a basis vector for the null space when it exists.

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Formula

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

The null space is the set of all vectors x that satisfy Ax = 0 (the homogeneous system). For a 2x2 matrix, compute the determinant first: if nonzero, the null space is just {0}. If zero, solve the system to find the null space basis vector.

Worked Example

A = [[1, 2], [2, 4]] Step 1: Determinant = (1)(4) - (2)(2) = 0 (singular matrix) Step 2: Solve Ax = 0: x1 + 2x2 = 0, so x1 = -2x2 Step 3: Choose x2 = 1, then x1 = -2 Step 4: Null space basis = {[-2, 1]} Step 5: Nullity = 1, Rank = 1 Result: N(A) = span{[-2, 1]}. The null space is a line through the origin.

What Is the Null Space of a Matrix?

The null space (also called the kernel) of a matrix A is the set of all vectors x such that Ax = 0. It is a fundamental subspace in linear algebra that reveals what information the matrix "loses" when applied as a transformation. The null space is always a subspace, meaning it contains the zero vector and is closed under addition and scalar multiplication.
  • A trivial null space (just the zero vector) means the matrix is invertible and the transformation is one-to-one
  • A non-trivial null space means the matrix maps some nonzero vectors to zero, losing information
  • The dimension of the null space is called the nullity
  • The Rank-Nullity Theorem states: rank + nullity = number of columns (always 2 for 2x2 matrices)
  • The null space is the orthogonal complement of the row space
  • In systems of equations, the null space describes all solutions to the homogeneous system Ax = 0

Understanding the null space is essential for solving systems of linear equations, understanding matrix invertibility, and working with linear transformations. A non-trivial null space means the system Ax = b either has no solution or infinitely many solutions.

You can also calculate changes using our Column Space Calculator, 2x2 Determinant Calculator, Matrix Rank Calculator or Eigenvalue Calculator.

Frequently Asked Questions

What does a trivial null space mean?

A trivial null space contains only the zero vector. This means the only solution to Ax = 0 is x = 0, the matrix is invertible (determinant is not zero), and the transformation is injective (one-to-one). No information is lost when the matrix is applied.

What is the Rank-Nullity Theorem?

For an m x n matrix, rank + nullity = n (number of columns). For a 2x2 matrix, rank + nullity = 2. So either rank = 2 and nullity = 0 (invertible), or rank = 1 and nullity = 1 (singular), or rank = 0 and nullity = 2 (zero matrix).

How is the null space related to eigenvalues?

The null space of (A - lambda I) is the eigenspace for eigenvalue lambda. Zero is an eigenvalue if and only if the null space of A itself is non-trivial. So checking the null space is equivalent to testing whether zero is an eigenvalue.

What is the kernel of a matrix?

Kernel is another name for null space. Both refer to the set of all vectors x satisfying Ax = 0. The term "kernel" is more common in abstract algebra and functional analysis, while "null space" is more common in matrix theory.

How does the null space relate to solutions of Ax = b?

If x_p is one particular solution to Ax = b, then the general solution is x_p + N(A), where N(A) is the null space. Every solution differs from every other solution by a null space vector. A trivial null space means a unique solution; a non-trivial null space means infinitely many.

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