Calculator
Enter one vector per line. Separate components with commas or spaces.
Example data table
| Vector Set | Dimension | Rank | Determinant | Conclusion |
|---|---|---|---|---|
| (1,0), (0,1) | 2 | 2 | 1 | Independent |
| (1,2), (2,4) | 2 | 1 | 0 | Dependent |
| (1,0,0), (0,1,0), (0,0,1) | 3 | 3 | 1 | Independent |
| (1,1,0), (0,1,1), (1,2,1) | 3 | 2 | 0 | Dependent |
Formula used
1) Matrix construction
Place the vectors as columns in a matrix A = [v1 v2 ... vn].
2) Rank criterion
The vectors are linearly independent exactly when rank(A) = n, where n is the number of vectors.
3) Homogeneous proof idea
Solve A·c = 0. If the only solution is c = 0, the vectors are independent. Any non-zero solution proves dependence.
4) Determinant shortcut
For a square matrix, det(A) ≠ 0 proves independence. If det(A) = 0, the square set is dependent.
How to use this calculator
- Enter one vector on each line.
- Separate components using commas or spaces.
- Choose the decimal precision you prefer.
- Adjust the tolerance when entries are very small.
- Submit the form to generate the proof.
- Read the rank, determinant, and RREF sections.
- Review the row operations when you need full detail.
- Download CSV or PDF for records or assignments.
FAQs
1) What does linear independence mean?
A set is linearly independent when no non-trivial combination of the vectors equals the zero vector. Only the zero coefficients may solve the homogeneous equation.
2) Why does rank decide independence?
Rank counts pivot columns after elimination. If every vector column gets a pivot, no vector is redundant. That gives independence.
3) When can determinant be used?
Determinant works only for square sets. That means the number of vectors must equal the dimension of each vector.
4) Can more vectors than dimension ever be independent?
No. In a space of dimension m, any set with more than m vectors must be dependent. The rank cannot exceed m.
5) What is the role of RREF here?
RREF exposes pivots and free variables clearly. It gives a stepwise proof that the homogeneous system has only trivial or non-trivial solutions.
6) Why is tolerance adjustable?
Very small decimal values can behave like zero numerically. Tolerance lets you control how strictly pivots are treated.
7) Does vector magnitude affect independence?
Not by itself. Large vectors can still be dependent, and small vectors can be independent. The structure of the set matters.
8) What does a dependence relation show?
It gives explicit coefficients, not all zero, whose combination equals the zero vector. That directly proves dependence.