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Inputs
Leave empty to auto-detect from inputs.
Enter expressions like 3x^2 - 2x + 7, x, or -x^3 + 0.5x.
Verdict
Example Data Table
These examples demonstrate independent polynomials. Click “Load example” above to insert.
| Label | Polynomial | Coefficients [1, x, x², …] |
|---|---|---|
| p₁ | 1 + x | [1, 1, 0] |
| p₂ | x + x^2 | [0, 1, 1] |
| p₃ | 1 + x^2 | [1, 0, 1] |
Coefficient Matrix (rows: degrees 0..d, columns: polynomials)
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RREF
Rank: –
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Formula Used
Given polynomials \(p_1(x),\dots,p_n(x)\), write each as a coefficient column vector \(v_i=[a_{0i},a_{1i},\dots,a_{di}]^\top\) where \(d\) is the maximum degree. Form the matrix \(A=[v_1\,v_2\,\dots\,v_n]\) of size \((d+1)\times n\).
The set \(\{p_i\}\) is linearly independent iff \(\mathrm{rank}(A)=n\). Otherwise, any nonzero vector \(c\) in the nullspace \(\mathcal{N}(A)\) gives a dependence relation \(\sum_{i=1}^n c_i p_i(x)=0\). We compute RREF to find rank, pivots, and a nullspace basis.
How to Use
- Choose the number of polynomials and optionally a degree limit.
- Enter each polynomial using x and powers like x^3.
- Click Compute to build the matrix and reduce to RREF.
- Read the rank, pivot columns, and a dependence relation if applicable.
- Export the results to CSV or PDF for your records.
FAQs
Use terms like 3x^2, -x, 7, joined by +/-.
The variable must be x. Spaces are fine.
No. The tool auto-detects the maximum degree among your polynomials unless you specify a cap.
We compute the rank of the coefficient matrix via RREF. Independence holds when the rank equals the number of polynomials (columns).
Yes. If dependent, we output one nontrivial nullspace vector \(c\) so that \(\sum c_i p_i(x)=0\).
Internally we use floating-point arithmetic with tolerances. Coefficients close to integers are rounded for readability.
The matrix always includes rows up to the maximum detected or chosen degree to avoid losing information.