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Column Space Calculator

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

About Column Space Calculator

The column space is the span of the columns, and its dimension is the rank. Row reduction identifies which columns are pivots; the corresponding original columns form a basis, not the reduced ones.

Formula

Column space

Col(A) = span of the columns; dim Col(A) = rank(A)

Worked example

With the default values below, Column Space Calculator returns Dimension of the column space of 2. Change any input above and every figure here is recalculated in your browser.

Inputs used

Matrix, row by row
1, 2, 1, 3, 4, 5, 5, 6, 9
Columns per rowaccepts 2–8
3

Results

Dimension of the column space
2
Basis columns from the original matrix
(1, 3, 5) (2, 4, 6)
Pivot column positions
1, 2
Nullity
1

Step by step

  1. 1Values used
  2. 2Column space
  3. 3Dimension of the column space= 2
  4. 4Basis columns from the original matrix= (1, 3, 5) (2, 4, 6)
  5. 5Pivot column positions= 1, 2
  6. 6Nullity= 1

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

Why use the original columns, not the reduced ones?

Row operations change the column space, though they preserve which columns are independent — so the pivot positions transfer but the entries do not.

Is the row rank the same as the column rank?

Yes, always. That equality is one of the fundamental results of linear algebra.

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