Matrix Settings
Columns independent if rank = n; rows independent if rank = m.
Enter Matrix A (m × n)
Numerical tolerance ε = 1e-10; small values are treated as zeros.
Results
Visualizations
Interactive heatmap of |A| through elimination steps and a pivot-column indicator chart. Use controls to step the algorithm and export PNGs.
0: Start – Original matrix
Step-by-Step Elimination Log
Example Data Table
| Example | Matrix A | Check | Rank | Decision | Action |
|---|---|---|---|---|---|
| Independent columns | [[1,0,0],[0,1,0],[0,0,1]] |
Columns | 3 | Independent | |
| Dependent columns | [[1,2,3],[2,4,6],[0,1,1]] |
Columns | 2 | Dependent | |
| Independent rows | [[1,2,0,1],[0,1,1,0]] |
Rows | 2 | Independent |
Formula Used
- Linear independence (columns): columns of A are independent ⇔ rank(A) = n.
- Linear independence (rows): rows are independent ⇔ rank(A) = m.
- Determinant test (square only): det(A) ≠ 0 ⇒ independent columns.
- RREF pivots: independent ⇔ every column (or row) has a pivot; free columns imply dependencies.
- Nullity: for columns, nullity = n − rank; a positive nullity gives a basis of dependencies. For rows, compute the left null space of A.
We use Gaussian elimination with partial pivoting and a tolerance ε = 1e-10.
How to Use
- Set matrix size (m rows, n columns).
- Choose whether to test columns or rows.
- Enter numbers or load an example.
- Click Check Independence to compute rank, pivots, determinant, and nullity.
- Review RREF table and visualize elimination steps and pivot columns.
- Download CSV or a PDF summary; export PNGs of charts.
For near-singular matrices, small numerical errors may affect decisions.
FAQs
A set of vectors is linearly independent if the only way to combine them to get the zero vector is the trivial combination where all coefficients are zero.
Yes. If m ≥ n and rank(A) = n, its n columns are independent even when m ≠ n.
For square matrices only: det(A) ≠ 0 implies independent columns and full rank. If det(A) = 0, at least one dependency exists.
RREF exposes pivot positions. Independent sets have pivots in every tested vector; free columns identify dependencies and allow building a null-space basis.
Nullity is the dimension of the solution space to Ax=0. If nullity > 0, the columns are dependent; the basis vectors give explicit linear relations.
Select “Rows”. Rank is unchanged, but we analyze Aᵀ for the left null space to interpret dependencies among rows.
We treat tiny numbers |x| < ε as zero. If results seem unstable, rescale inputs or use rational arithmetic for exact work.