Enter Vector Data
The page stays single-column overall, while the calculator inputs use 3 columns on large screens, 2 on medium, and 1 on mobile.
Example Data Table
| Dimension | v1 | v2 | v3 | Target | Rank | Determinant | Basis? | Coordinates of Target |
|---|---|---|---|---|---|---|---|---|
| R^3 | (1, 0, 1) | (0, 1, 1) | (1, 1, 0) | (2, 3, 3) | 3 | -2 | Yes | (1, 2, 1) |
This sample shows a full basis in R^3 because the vectors are independent and span the whole space.
Formula Used
1) Basis test
A set of vectors forms a basis for R^n when it is linearly independent and spans R^n. In matrix form, use the vectors as columns of A.
2) Rank and pivot columns
Row reduction transforms A into reduced row echelon form. The number of pivots gives the rank. Pivot columns identify a basis for the span.
3) Determinant rule for square sets
If A is square, then det(A) ≠ 0 means the columns are independent and span the whole space, so they form a basis.
4) Coordinate recovery
To express a target vector b in the basis, solve A c = b. The solution vector c contains the coordinates of b relative to that basis.
How to Use This Calculator
- Choose the vector dimension and the number of supplied vectors.
- Enter each vector component in the responsive input cards.
- Fill the target vector if you want span and coordinate analysis.
- Press Submit and Calculate.
- Review the classification, rank, determinant, pivot basis vectors, and coordinate results above the form.
- Use the CSV and PDF buttons to save your findings.
Frequently Asked Questions
1) What does this calculator determine first?
It builds a matrix from your vectors, reduces it, finds the rank, and then checks independence, spanning, basis status, and target vector membership.
2) Why are pivot columns important?
Pivot columns show which original vectors are essential. Those vectors form a basis for the span generated by the entire supplied set.
3) Can more vectors than the dimension form a basis?
No. In R^n, any basis has exactly n vectors. Extra vectors force dependence, although they may still span the same space.
4) Does a nonzero determinant always guarantee a basis?
Yes, but only for square matrices. A nonzero determinant means the columns are independent and span the full space.
5) What happens if the target vector is in the span but the set is not a basis?
The target can still be represented, but the representation may not be unique. Unique coordinates require a genuine basis.
6) Why might the graph show only three components?
For dimensions above three, the plot uses the first three components so the vectors can still be visualized in a practical way.
7) Can the calculator extract a basis from a dependent set?
Yes. The pivot columns identify a smaller subset of the original vectors that still spans the same column space.
8) When should I rely on rank instead of determinant?
Use rank for any rectangular or square set. Determinant works only for square matrices and cannot analyze general rectangular systems.