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Enter your values, then click Calculate result.How this calculator helps
This basis calculator identifies independent directions associated with a real matrix. It reports the row-reduced matrix and three different space bases instead of treating the word basis as one interchangeable result. Use it to check a linear algebra exercise, inspect dependence in a small data matrix or verify a homogeneous system. The algorithm uses numerical floating-point elimination, so its disclosed rank tolerance matters when entries are nearly dependent.
How to use it
- 1
Paste the rectangular real matrix with one row per line.
- 2
Verify every row has the same number of columns and no symbolic fractions.
- 3
Click Calculate result to inspect RREF, pivots and the three space bases.
- 4
Multiply each null-space vector by the original matrix and review numerical tolerance.
Formula and methodology
Row-reduce A with partial pivoting. Pivot count is rank. Original pivot columns form a column-space basis. Nonzero RREF rows form a row-space basis. Free variables generate a basis of Ax = 0.
The calculator applies the displayed arithmetic to the values entered on this device. It does not silently load a local tax rate, currency conversion or commercial assumption.
Worked calculation example
For A with rows [1,2,3] and [2,4,6], the second row is twice the first. Its RREF is [1,2,3] followed by [0,0,0]. Rank is 1 and nullity is 2. The original first column [1,2] is a column-space basis. Setting the second free variable to one gives a null vector [−2,1,0]; setting the third to one gives [−3,0,1]. Multiplying A by either vector produces the zero vector.
How to interpret your result
Read rank together with nullity and the actual vector lengths. The RREF supplies the pivot equations; the original-column basis preserves the original output directions. If a data matrix is badly scaled or nearly dependent, the result is a numerical classification under a stated tolerance, rather than proof of exact symbolic dependence. Keep the original matrix and threshold with any reported basis.
For different inputs or formulas, use Matrix Multiplication Calculator; Substitution Calculator; Matrix Inverse Calculator.
Related questions this calculator covers
- basis calculator
- matrix basis calculator
- null space basis calculator
- rref rank calculator
Scenario comparison
| Scenario | What it shows |
|---|---|
| Dependent rows | [1,2,3] and [2,4,6] have rank one. |
| Identity matrix | every column is a pivot and nullity is zero. |
| Zero matrix | rank is zero and the standard coordinate vectors span its null space. |
Common mistakes to avoid
- Using reduced columns as a basis of the original column space.
- Listing the zero vector as a basis vector.
- Treating a numerical rank decision as exact rational algebra.
For the worked example, verify A[−2,1,0]ᵀ and A[−3,0,1]ᵀ both vanish. Count basis vectors and check rank plus nullity equals the number of columns. Test a rectangular zero matrix and an identity matrix. Scale all original entries by a nonzero moderate factor and confirm the pivot count and null-space directions remain consistent within the relative tolerance.
Authoritative reference. Reviewed on 6 October 2026 for the specific disclosed method or measurement context. The arithmetic is independently implemented. A linked reference does not certify an individual result or extend the worksheet beyond its stated scope.What can affect the result?
Enter a rectangular matrix
Paste one row per line, with entries separated by spaces or commas. Every row must have the same number of columns. The supported size is one through eight rows and one through eight columns, with real finite entries bounded in magnitude by one million. Fractions such as 1/3 are not parsed as symbolic expressions; enter an appropriate decimal approximation and recognize its precision. Labels, complex numbers and an empty row are rejected rather than silently removed.
Rank counts pivot directions
Partial pivoting chooses the largest available entry in the current column before normalizing the pivot row and eliminating that column elsewhere. The count of accepted pivots is the numerical rank. For an m-by-n matrix, rank cannot exceed either m or n. Pivot columns describe independent input columns under the stated tolerance. Their positions are displayed using one-based numbering, so column one means the first visible column, not a programming index.
Use original columns for column space
Row operations preserve the relationships between columns but generally change the column vectors themselves. The column-space basis therefore takes the pivot column positions from RREF and selects those columns from the original matrix. Copying pivot columns directly out of RREF would describe a transformed column space rather than necessarily the original one. The output lists each original column vector separately and retains its original number of row coordinates.
Nonzero reduced rows form a row basis
Elementary row operations preserve the span of the rows. The independent nonzero RREF rows can therefore serve as a row-space basis. These vectors have one coordinate per original matrix column, which differs from the dimension of an original column vector when the matrix is rectangular. A basis is not unique: another independent set can span the same space. A different valid answer does not have to match the exact displayed vector entries.
Null space solves the homogeneous system
Null vectors have n coordinates and satisfy Ax = 0. For each free column, the worksheet sets its free variable to one and the other free variables to zero, then reads the required pivot-variable values from RREF. These generated vectors span the numerical null space. If every column is a pivot, the null space contains only the zero vector and its basis is empty; the zero vector itself is not an independent basis vector.
Tolerance determines numerical dependence
The pivot threshold is 10^−10 times the largest absolute entry in the original matrix. Entries below that scale can be treated as numerically zero when deciding rank. Rows with no accepted pivot are treated as numerical zero rows. Normalized display values smaller than 10^−10 are also shown as zero. This convention is not exact rational algebra and is not a measurement uncertainty model. Near-dependent matrices may need higher precision, exact arithmetic or a singular-value method. Review the original scale when a tiny perturbation changes a pivot decision.
Check dimensions and products
A column-space basis contains rank vectors of length m; a row-space basis contains rank vectors of length n. A null-space basis contains n minus rank vectors, each of length n. These counts provide quick consistency checks. Multiply every displayed null vector by the original matrix, not the reduced matrix alone, to inspect residuals. For floating-point input, small residuals should be interpreted relative to the input scale rather than demanding literal binary zero.
Privacy and browser processing
Values entered on this page are processed in the current browser session. SolvePilot does not require an account and does not receive the values entered into the calculator. Refreshing or closing the page clears the working values unless the browser itself restores a previous session. Avoid entering identifying or account information because the calculation needs summary values only.
Accuracy and verification
Accuracy depends first on input quality. Confirm definitions, scales, dates and source information before entering a value. Keep an independent record of any result used for planning because this page does not create an official statement or retain a calculation history.
Limits of this estimate
Using reduced columns as a basis of the original column space. Read rank together with nullity and the actual vector lengths. The RREF supplies the pivot equations; the original-column basis preserves the original output directions. If a data matrix is badly scaled or nearly dependent, the result is a numerical classification under a stated tolerance, rather than proof of exact symbolic dependence. Keep the original matrix and threshold with any reported basis. Entries remain on this device for the calculation and are not submitted to a calculation server. Browser floating-point arithmetic and displayed rounding do not establish real-world measurement certainty.
Sources and review information
Frequently asked questions
Can a basis differ from my textbook answer?＋
Yes. A space can have many valid bases; compare span and independence.
Are column basis vectors taken from RREF?＋
No. Pivot positions select vectors from the original matrix.
What does an empty null-space basis mean?＋
Only the zero vector solves Ax = 0 under the numerical rank decision.
Can I enter exact fractions?＋
No. This interface accepts real decimal numbers and uses floating-point arithmetic.
How large can the matrix be?＋
The supported bound is eight rows by eight columns.