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The row space of a matrix is the set of all possible linear combinations of its rows, and — unlike the column space — a basis for it can be read directly off the row echelon form, since row operations never change the row space.
This calculator finds a basis for a matrix's row space and its rank.
How it works
Enter a matrix of any size. The calculator reduces it to row echelon form, then takes every non-zero row of that reduced form directly as a basis vector for the row space.
- Click Calculate to see your results.
Examples
[[1,2],[2,4]]
Since the second row is exactly twice the first, the row echelon form has only one non-zero row — the row space basis is that row, (1, 2), giving a rank of 1.
Who should use it
- Finding a basis for a matrix's row space in a linear algebra course.
- Checking the rank of a matrix via its row space dimension.
Industry applications
- Linear algebra and mathematics education
- Engineering and data science coursework
Advantages
- Correctly reads the basis directly from the row-reduced form, which is valid for row space (unlike column space).
- Works for rectangular matrices, not just square ones.
Limitations
- Reports a standard basis derived from row reduction — a different (but equally valid) basis could also be constructed.
Common mistakes to avoid
- Assuming row space and column space are the same subspace just because they share the same dimension (rank) — they're generally different subspaces entirely.
- Using the ORIGINAL matrix's rows instead of the reduced form's rows — unlike column space, row space basis vectors should come from the reduced (echelon) form.
Best practices
- Remember the key asymmetry: row space basis comes from the REDUCED form's rows, while column space basis comes from the ORIGINAL matrix's columns — mixing these up is a common error.
Tips
- Pair this with the site's Column Space Basis Calculator on the same matrix — both report the same rank, a good way to double-check your row reduction work.