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Column Space Calculator

MathAlgebra Tools

Up to 10 rows and 10 columns. Decimals and fractions such as -3/4 are accepted.

Dimension of the column space (rank) —
Matrix size (rows x columns)
—
Pivot columns (basis)
—
Nullity (dim Nul A)
—
Full rank
—
Columns span R^m
—

The column space of a matrix A is the set of all linear combinations of its columns. Row-reduce A, find the pivot columns, and take those columns from the original A as the basis. For the matrix with rows 1 2 3, 4 5 6, 7 8 9, columns a1 and a2 are pivots, so Col A is a plane in R^3 with dimension 2.

About this tool

Students working through a linear algebra course meet the column space early, and checking a hand reduction is tedious: one arithmetic slip moves a pivot and changes the whole answer. Paste a matrix with one row per line and the tool row-reduces it, names the pivot columns, lists the basis vectors taken from your original matrix, and reports the rank, the nullity and whether the columns span all of R^m. The working lists each row exchange, scaling and pivot decision, followed by the reduced echelon form. Entries can be integers, decimals or fractions such as 5/3. One limit: arithmetic is in floating point with a small tolerance, so a matrix whose entries differ by around one part in a billion can be judged dependent when exact arithmetic would say otherwise.

How to use it

  1. Type or paste the matrix

    Put each row on its own line and separate entries with spaces or commas. Fractions like -3/4 are fine.

  2. Read the rank

    The headline number is the dimension of the column space, which equals the number of pivot columns.

  3. Take the basis from the table

    Each row of the table is one pivot column copied from your original matrix, not from the reduced form.

  4. Check the working

    The steps show the pivot found in each column and the final reduced row echelon form, so you can compare them with your own reduction.

Examples

The 1-to-9 matrix

Matrix A
1 2 3 4 5 6 7 8 9

Result Dimension of the column space (rank): 2
Matrix size (rows x columns): 3 x 3
Pivot columns (basis): a1, a2
Nullity (dim Nul A): 1
Full rank: No
Columns span R^m: No, a 2-dimensional subspace of R^3

  1. Swap R1 and R3.
  2. Column 1 is a pivot column: scale R1 by 1/7 and clear the column.
  3. Swap R2 and R3.
  4. Column 2 is a pivot column: scale R2 by 1/0.8571 and clear the column.
  5. Column 3: no pivot below row 2, so it depends on earlier columns.
  6. RREF = [1, 0, -1] [0, 1, 2] [0, 0, 0]

The third column equals twice the second minus the first, so only columns 1 and 2 are needed: the column space is a plane in R^3.

Two parallel columns

Matrix A
1, 2 2, 4 3, 6

Result Dimension of the column space (rank): 1
Matrix size (rows x columns): 3 x 2
Pivot columns (basis): a1
Nullity (dim Nul A): 1
Full rank: No
Columns span R^m: No, a 1-dimensional subspace of R^3

  1. Swap R1 and R3.
  2. Column 1 is a pivot column: scale R1 by 1/3 and clear the column.
  3. Column 2: no pivot below row 1, so it depends on earlier columns.
  4. RREF = [1, 2] [0, 0] [0, 0]
  5. Basis of Col A = original columns a1; dim Col A = rank = 1.

Column 2 is twice column 1, so the column space is the line through (1, 2, 3) in R^3.

A 2 x 4 matrix with fractions

Matrix A
1 0 2 1 0 1 3 -1/2

Result Dimension of the column space (rank): 2
Matrix size (rows x columns): 2 x 4
Pivot columns (basis): a1, a2
Nullity (dim Nul A): 2
Full rank: Yes
Columns span R^m: Yes, all of R^2

  1. Column 1 is a pivot column: scale R1 by 1/1 and clear the column.
  2. Column 2 is a pivot column: scale R2 by 1/1 and clear the column.
  3. RREF = [1, 0, 2, 1] [0, 1, 3, -0.5]
  4. Basis of Col A = original columns a1, a2; dim Col A = rank = 2.

The first two columns are the standard basis of R^2, so the four columns span all of R^2 and two columns are free.

How it is calculated

Col A = span{a_j : j is a pivot column of rref(A)}, dim Col A = rank A = n - dim Nul A

A
the m x n matrix you enter
a_j
column j of the original matrix A
rref(A)
the reduced row echelon form of A
n
number of columns; dim Nul A is the nullity

Row operations change the columns themselves but keep every linear relation among them. So a column that has no pivot in the echelon form is a combination of the pivot columns to its left, in A as well as in rref(A). The pivot columns of the original matrix are therefore linearly independent and span Col A, which makes them a basis. The tool uses Gauss-Jordan elimination with partial pivoting and treats any value smaller than one billionth of the largest entry as zero.

When not to use it

  • It only accepts numbers, so it cannot handle a matrix with a parameter such as k.
  • Irrational entries are rounded to decimals before elimination.
  • For badly conditioned matrices, a numerical rank from singular values is more reliable.

Common mistakes

  • Many students list the pivot columns of the reduced form as the basis.
  • Those are usually unit vectors and generally do not lie in Col A.
  • The column space lives in R^m, while the row space lives in R^n.
  • Check that every row has the same number of entries.

Frequently asked questions

Why is the basis taken from the original matrix and not the reduced one?

Row operations mix rows, which changes the column vectors themselves. The pivot positions tell you which columns are independent, but the vectors that actually span Col A are the original columns at those positions. The reduced columns span a different space in general.

How are rank and nullity related?

For an m x n matrix, rank plus nullity equals n, the number of columns. Every column is either a pivot column, adding one to the rank, or a free column, adding one to the dimension of the null space. A 3 x 4 matrix of rank 2 therefore has nullity 2.

When do the columns span all of R^m?

Exactly when there is a pivot in every row, so the rank equals m, the number of rows. A matrix with more rows than columns can never do this, because rank cannot exceed the number of columns.

Is the basis unique?

No. Any set of rank-many independent vectors inside Col A is a basis. The pivot-column basis is a standard choice because it uses vectors you already have. Another common choice is the nonzero rows of the reduced form of A transposed, which gives a different but equally valid basis.

What does the column space mean for solving Ax = b?

The system Ax = b has a solution exactly when b lies in Col A. If the rank equals the number of rows, every b works. Otherwise only vectors in the lower-dimensional subspace spanned by the pivot columns can be reached.

Which input formats are accepted?

One row per line, or rows separated by semicolons, with entries split by spaces or commas. Integers, decimals, scientific notation such as 2e-3, and simple fractions such as 7/2 are accepted, up to 10 rows and 10 columns.

Cite this tool

Using this tool in an assignment, article or report? Copy a citation, or a link for your website.

APA

Tools Directory Editorial Team. (2026, October 25). Column Space Calculator. Tools Directory. Retrieved October 26, 2026, from https://tools.milunir.com/math/algebra-tools/column-space-calculator/

MLA

Tools Directory Editorial Team. “Column Space Calculator.” Tools Directory, 25 Oct. 2026, https://tools.milunir.com/math/algebra-tools/column-space-calculator/. Accessed 26 Oct. 2026.

Chicago

Tools Directory Editorial Team. “Column Space Calculator.” Tools Directory. Last modified October 25, 2026. Accessed October 26, 2026. https://tools.milunir.com/math/algebra-tools/column-space-calculator/.

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