Turn vectors into a reliable basis in minutes. See rank and pivot vectors instantly here. Download CSV or PDF to share your results easily.
| Input vectors | Expected basis vectors | Notes |
|---|---|---|
| v1=[1,2,3], v2=[2,4,6], v3=[1,0,1], v4=[0,1,1] | [1,2,3], [1,0,1], [0,1,1] | v2 is dependent on v1; rank becomes 3. |
Let your vectors be v1, v2, …, vm in R^n. Form the column matrix A = [v1 v2 … vm] (an n×m matrix).
A basis is the smallest set of vectors that can rebuild every vector in a span. In modelling, a compact basis reduces parameters, improves interpretability, and exposes redundancy. For an m-vector input set, the calculator reports rank r and returns r basis vectors, giving an immediate measure of compression from m to r.
The tool forms a matrix whose columns are your vectors, then applies row-reduction to find pivot positions. Each pivot marks an original column that contributes new directional information. The number of pivots equals rank, and rank equals the dimension of the column space. If you enter five 4D vectors and rank is 3, then the span is 3-dimensional, not 4. This reveals independent axes.
Linear independence is confirmed when rank equals the number of input vectors. If rank is smaller, at least one vector is a combination of earlier pivots. The reduced form also indicates which vectors are redundant: non‑pivot columns are dependent. This matters in feature engineering, where dependent features inflate variance without adding signal. In geometry, dependence means the vectors lie in the same plane or subspace.
With decimals, tiny rounding noise can create near‑zero pivots. The tolerance setting treats values below a threshold as zero, stabilizing rank decisions. For example, if a pivot candidate is 0.0000004 and tolerance is 1e-6, it is ignored. Consistent scaling (similar magnitudes across components) also improves robustness. When vectors vary by 10^6, rescale before evaluation for clearer pivots.
After computing, export the basis, rank, and diagnostics to CSV for spreadsheets or code review. The PDF export is useful for reports, assignments, and audit trails because it captures the chosen tolerance and the selected pivot indices. Re-run with different vector orders to confirm the same rank and to study alternate valid bases. If rank stays constant, your span remains unchanged when the basis changes.
It returns the rank of the input set and a basis made from the original vectors. It also lists pivot indices, flags independence, and shows diagnostic matrices to justify the selection.
The calculator row-reduces the matrix of input vectors and chooses the original columns that correspond to pivot positions. Those pivot columns form a valid basis for the column space.
Rank is the number of independent directions in your vectors. It equals the dimension of the span. If rank is smaller than the number of vectors, some vectors add no new information.
A span can have many valid bases. Changing vector order or swapping dependent vectors may change which pivot columns are chosen, while the rank and the spanned space remain the same.
Use a small value when inputs are clean integers. For decimal data, start around 1e-9 to 1e-6 depending on scale. If results seem unstable, rescale vectors and increase tolerance slightly.
All vectors must share the same dimension. If a line has fewer numbers, pad or correct it before submitting. Mixed dimensions cannot form a single matrix for row-reduction.
Important Note: All the Calculators listed in this site are for educational purpose only and we do not guarentee the accuracy of results. Please do consult with other sources as well.