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Null Space Calculator

Work out null space instantly with clear inputs, formula shown and shareable results.

About Null Space Calculator

The null space contains every vector A maps to zero. Row reducing exposes the free columns, and setting each free variable to 1 in turn generates a basis. Rank plus nullity always equals the number of columns.

Formula

Rank-nullity theorem

rank(A) + nullity(A) = number of columns

Worked example

With the default values below, Null Space Calculator returns Kernel dimension (nullity) of 2. Change any input above and every figure here is recalculated in your browser.

Inputs used

Matrix, row by row
1, 2, 3, 2, 4, 6
Columns per rowaccepts 2–8
3

Results

Kernel dimension (nullity)
2
Basis of the kernel
(-2, 1, 0) (-3, 0, 1)
Rank
1
Free columns
2, 3

Step by step

  1. 1Values used
  2. 2Rank-nullity theorem
  3. 3Kernel dimension (nullity)= 2
  4. 4Basis of the kernel= (-2, 1, 0) (-3, 0, 1)
  5. 5Rank= 1
  6. 6Free columns= 2, 3

Assumptions and limits

  • The result is derived only from the 2 inputs listed above. Anything not among them is not part of the calculation.
  • In particular, this calculator does not take inflation, tax, fees or charges or compounding frequency into account.
  • Figures are rounded for display to whole numbers, so totals built from several rounded parts can differ slightly from the unrounded arithmetic.

How these calculators are built and tested is described on the Methodology page. If a figure looks wrong, please report it.

Frequently Asked Questions

For [[1,2,3],[2,4,6]]?

The second row is twice the first, so the rank is 1 and the nullity is 2.

When is the null space trivial?

When the rank equals the number of columns, meaning the columns are linearly independent and only zero maps to zero.

Related Calculators

Null Space Calculator – Calculate Kernel dimension (nullity)