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Null Space Calculator

Find all vectors that a 2 by 2 matrix sends to zero, then see the row steps and the basis for the null space during introductory verification.

Matrix [[a, b], [c, d]]

Result

(-2, 1)

The null space of [[1, 2], [2, 4]] is spanned by (-2, 1)

Quick Answer

A null space calculator solves the homogeneous system Ax = 0. For A = [[1, 2], [2, 4]], the rows are linearly dependent. Let x2 = t, giving x1 = -2t and x = t(-2, 1). Thus (-2, 1) is a basis vector. When det(A) is nonzero, invertibility leaves only x = (0, 0).

How A 2 By 2 Null Space Calculator Finds A Basis Vector

The null space is all input vectors that a matrix sends to zero. This tool works with a 2 by 2 matrix. The tool writes two equations from Ax = 0 and cuts them to the key row. If det(A) is not zero, the matrix has an inverse. Only the zero vector can work. If det(A) is zero, one row may be a scaled copy of the other. One value is then free. Set that value to t and solve for the other one. The answer is t times a nonzero vector. That vector spans a line through the origin. The small matrix keeps each row step easy to see. A final check multiplies A by the basis vector and must give (0, 0).

The Math Behind A 2 By 2 Matrix Null Space Basis Vector

To find the basis of a null space, the calculator solves the homogeneous system Ax = 0 using row reduction and free variables.

  • Solve Ax = 0 for A = [[a, b], [c, d]] and x = (x1, x2).
  • Determinant: det(A) = ad − bc.
  • If det(A) = 0, find a parameter t and express the basis vectors.
  • If det(A) is not zero, the only solution is the zero vector.

a, b, c, d: the entries of the 2 by 2 matrix.

det(A): the determinant of the matrix.

A basis lists independent direction vectors whose linear combinations produce every null-space vector.

How To Use A Null Space Calculator In 5 Clear Steps

Inputs

  • a and b: enter the first row of the matrix.
  • c and d: enter the second row of the matrix.

Steps

  1. Enter all four matrix entries.
  2. Calculate the determinant.
  3. Read the reduced homogeneous equation.
  4. Identify the free parameter when the determinant is zero.
  5. Verify each basis vector with A times v.

Null Space Basis Example For A Singular Matrix By Hand

Matrix A = [[1, 2], [2, 4]].

  1. Find the determinant: det(A) = (1 × 4) − (2 × 2) = 0.
  2. Write Ax = 0: x1 + 2x2 = 0 and 2x1 + 4x2 = 0.
  3. Keep only the independent equation: x1 + 2x2 = 0.
  4. Let free parameter x2 = t. Then x1 = −2t.
  5. Write solution vector: x = (−2t, t) = t(−2, 1).

The basis for Null(A) is span{(−2, 1)}. Exact arithmetic is used, so no rounding is needed. Checked: A(−2, 1) = (0, 0).

Is A Small Matrix Null Space Tool Right For Classwork?

Use this tool for a first null-space task with a 2 by 2 matrix. The clear row work also helps you test a vector that you found by hand.

For nearby matrix algebra, determinant, and other linear systems tools, see the math calculators hub and look up concepts in the glossary.

Assumptions

  • The matrix is 2 by 2.
  • Entries are real numbers.
  • The task is Ax = 0.
  • Any nonzero scaling gives an equivalent basis vector.

Limitations

  • The calculator does not handle matrices larger than 2 by 2.
  • Larger matrices are planned, not supported now.
  • Decimal inputs may hide exact dependence because of rounding.
  • A basis is not unique, since scaled direction vectors span the same line.

In Practice

The zero vector belongs to every null space but cannot serve as a basis vector. A valid basis requires nonzero, linearly independent vectors that span the full solution set. Use the free-variable form, then verify each direction by matrix multiplication.

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Related Guides

Null Space Calculator Questions And Worked Answers

What Does A Null Space Basis Represent?

A null-space basis gives the main directions that the matrix sends to zero. Each vector in the null space is a scaled sum of those basis vectors. For a rank-one 2 by 2 matrix, one nonzero vector spans the line.

Why Does A Nonzero Determinant Give Only Zero?

A nonzero determinant means a 2 by 2 matrix is invertible over real numbers. Apply the inverse to both sides of Ax = 0. The result is x = 0, so no nonzero vector can be in the null space.

Is The Null Space The Same As The Kernel?

Null space and kernel are two names for the same solution set in matrix algebra. Both mean all vectors x that solve Ax = 0. Most linear algebra texts use the terms in the same way.

How Can I Check A Null Space Vector By Hand?

Check a null-space vector by multiplying the matrix by that proposed vector. A valid vector must give zero in both rows. For (-2, 1), the two row checks are (1 × -2) + (2 × 1) = 0 and (2 × -2) + (4 × 1) = 0.

Sources

Last updated: . Reviewed for accuracy against the formula shown above.