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The column space of a matrix is the set of all possible linear combinations of its columns, and a basis for it can be found by identifying which of the ORIGINAL matrix's columns are linearly independent — distinct from the Null Space calculator, which instead uses the free (non-pivot) columns to describe solutions to Ax = 0.
This calculator finds a basis for a matrix's column space and its rank.
How it works
Enter a matrix of any size. The calculator reduces it to row echelon form to identify which columns contain a pivot, then takes those SAME column positions from the original (unreduced) matrix — those columns are guaranteed to be linearly independent and span the column space.
- Click Calculate to see your results.
Examples
[[1,2],[2,4]]
Since the second column is exactly twice the first, only the first column is a pivot column — the column space basis is just that one original column, (1, 2), giving a rank of 1.
Who should use it
- Finding a basis for a matrix's column space in a linear algebra course.
- Checking the rank and independence of a matrix's columns.
Industry applications
- Linear algebra and mathematics education
- Engineering and data science coursework (column space relates directly to the range of a linear transformation)
Advantages
- Correctly uses the original matrix's columns, not the RREF's columns, avoiding a common mistake.
- Works for rectangular matrices, not just square ones.
Limitations
- Reports a standard basis derived from pivot columns — a different (but equally valid) basis could also be constructed from other independent column combinations.
Common mistakes to avoid
- Reporting the reduced row echelon form's columns as the column space basis — row reduction doesn't preserve the column space, only which columns are pivots.
- Confusing column space (spanned by columns) with row space (spanned by rows) — they have the same dimension (the rank) but are generally different subspaces.
Best practices
- Remember that only the SAME pivot column positions taken from the ORIGINAL matrix form a valid column space basis — never use the reduced form's column values directly.
Tips
- The rank-nullity theorem connects this tool directly to the site's Null Space Calculator: for an n-column matrix, rank (column space dimension) plus nullity (null space dimension) always equals n.