Rows separated by new lines or semicolons. Values separated by spaces or commas. Exact mode supports integers, fractions like 3/5, and simple decimals.
Example matrix A (3×4) and its kernel and image summary.
| Row | c1 | c2 | c3 | c4 |
|---|---|---|---|---|
| r1 | 1 | 2 | 1 | 0 |
| r2 | 2 | 4 | 2 | 0 |
| r3 | 0 | 0 | 1 | 1 |
Kernel (null space): Ker(A) = { x ∈ ℝⁿ : A x = 0 }.
Image (column space): Im(A) = { A x : x ∈ ℝⁿ }, equal to the span of A’s pivot columns.
Rank–Nullity: rank(A) + nullity(A) = n, where n is the number of columns.
The calculator computes the RREF of A using Gauss–Jordan elimination. Pivot columns are the columns containing leading 1s in RREF. Free variables generate the kernel basis by setting one free variable to 1 and the others to 0, then solving for pivot variables from the RREF equations.
- Paste your matrix using spaces or commas between values.
- Choose Exact for fractions, or Decimal for floats.
- Set decimal places; adjust tolerance for noisy decimals.
- Click Submit to compute RREF, kernel, and image.
- Use CSV or PDF buttons to export your results.
Interpreting Kernel and Image
The kernel collects every vector x that solves Ax = 0, revealing directions crushed to zero by the transformation. The image is the set of reachable outputs Ax, which equals the span of pivot columns. Together they describe what information is lost and what output space is covered. In applications, a nontrivial kernel signals redundancy or constraints, while a small image signals limited output degrees of freedom.
Why RREF Drives the Results
Row-reducing to reduced row echelon form keeps the solution set unchanged while exposing pivot positions. Pivot columns identify a basis for the image, because those original columns generate all attainable outputs. Free columns correspond to free variables, which generate kernel basis vectors by assigning one free variable to 1 and others to 0. Each resulting basis vector satisfies A v = 0, and the set is linearly independent by construction.
Exact vs Decimal Workflows
Exact mode treats inputs as integers or fractions, so the basis vectors are rational and reproducible, ideal for symbolic homework checks. Decimal mode is faster for measured data, where rounding is unavoidable. The tolerance setting decides when a tiny value is treated as zero, helping stabilize pivots in noisy matrices. Scale inputs to reduce error.
Rank, Nullity, and Dimension Checks
Rank is the number of pivots, so it matches the dimension of the image. Nullity is the number of free variables, so it matches the dimension of the kernel. The rank-nullity identity rank(A) + nullity(A) = n provides a quick integrity check: if it fails, the input parsing or numerical tolerance likely needs adjustment. For m×n matrices, rank cannot exceed min(m, n); if you see a higher value, recheck row lengths and separators.
Exporting Results for Reports
After computation, you can export kernel and image bases as CSV for spreadsheets or as a compact PDF for sharing. CSV is best when you want to plot components, compare runs, or feed a downstream script. PDF is useful for clean documentation of rank, nullity, pivots, and basis vectors. When presenting results, report the pivot column indices and the chosen basis, since different but equivalent bases can span the same subspace.