Calculator
Example Data Table
| Function set | Functions | Suggested x0 | Expected result |
|---|---|---|---|
| Polynomial basis | 1, x, x^2 | 1 | Independent |
| Repeated relation | x, 2*x, 3*x | 1 | Dependent |
| Trigonometric pair | sin(x), cos(x) | 0 | Independent |
| Exponential pair | exp(x), exp(2*x) | 1 | Independent |
Formula Used
Linear Dependence Test
Functions f1, f2, ..., fn are linearly dependent when constants c1, c2, ..., cn exist, not all zero, such that:
c1 f1(x) + c2 f2(x) + ... + cn fn(x) = 0
Wronskian Formula
The Wronskian uses derivatives up to order n - 1:
W(f1,...,fn)(x) = det [ f_j^(i-1)(x) ]
If the Wronskian is nonzero at a point, the functions are linearly independent on a suitable interval.
Sample Matrix Formula
The calculator also builds a value matrix:
Aij = fj(xi)
If this matrix has full column rank, the sampled evidence supports linear independence.
How to Use This Calculator
- Enter one function of x on each line.
- Choose an interval where all functions are defined.
- Enter custom points, or let the calculator generate them.
- Set x0 for the Wronskian test.
- Adjust tolerance when values are very small.
- Press calculate and read the result above the form.
- Download CSV or PDF files for records.
Article
Linear Independence in Study Work
Linear independence tells whether a list of functions carries repeated information. A set is independent when no nonzero constants can combine the functions into the zero function. This idea appears in differential equations, vector spaces, signals, and approximation methods. It helps decide whether a proposed basis is useful.
Why This Calculator Helps
Manual testing can be slow. You may need a Wronskian, a sampled value matrix, or both. This tool accepts several functions of x. It evaluates them at selected points. It also estimates derivative rows for a Wronskian matrix. Then it reports rank and determinant evidence.
The result should be read with care. A nonzero Wronskian at one point proves independence for differentiable functions on a suitable interval. A full rank sampled matrix also gives strong evidence. Zero results can mean dependence. They can also mean poor points, rounding, or functions outside the method limits.
Choosing Good Inputs
Use simple expressions first. Write one function per line. Choose an interval where the functions are defined. Avoid points where logarithms, roots, or divisions fail. For trigonometric functions, try more than one point. Periodic functions can sometimes look dependent at unlucky samples.
Tolerance controls how close to zero a value may be. A smaller tolerance is stricter. A larger tolerance is safer for noisy decimal data. Step size controls numerical derivatives. Very small steps can amplify rounding errors. Very large steps can hide local behavior.
Best Practice
Compare both methods whenever possible. Check the sampled rank over several points. Then inspect the Wronskian near a regular point. If both support the same conclusion, confidence improves. For formal homework, use the formulas shown here as your written proof guide.
This calculator is designed for learning and review. It is not a symbolic proof engine. It works best for standard expressions, clean domains, and moderate function counts. Use the exported report to document values, determinants, and settings. Keep notes about any domain restriction. A clear setup makes the conclusion easier to defend.
When results disagree, change the sample points and try again. Record every setting. This habit separates real dependence from numerical coincidence. It also teaches how function spaces behave beyond plain vectors. Save exports for later comparison.
FAQs
What is linear independence of functions?
Functions are linearly independent when no nonzero constant combination makes the zero function over the chosen domain.
What does a nonzero Wronskian mean?
A nonzero Wronskian at one valid point proves linear independence for differentiable functions on a suitable interval.
Does a zero Wronskian always mean dependence?
No. A zero Wronskian may be inconclusive in some settings. Check definitions, domains, and other proof methods.
Why does the calculator use sample points?
Sample points create a value matrix. Full rank in that matrix gives useful numerical evidence for independence.
How many functions can I test?
This page limits testing to six functions. That keeps derivative estimates and determinant calculations stable.
Which function syntax is supported?
Use x, numbers, pi, e, +, -, *, /, ^, and functions like sin, cos, exp, ln, log, sqrt, and abs.
Why did I get a domain error?
A sample point may be outside the function domain. Check logarithms, square roots, and division expressions.
Can I export the result?
Yes. Use the CSV button for spreadsheet data. Use the PDF button for a readable report summary.