The Wronskian Method

Linear Algebra (Matrices) · Axiom Academy

A determinant test that decides whether functions, viewed as vectors, are linearly independent. Functions are linearly independent when the only way a combination vanishes for every x is the trivial one — all coefficients zero: To test this, build the Wronskian W(x) : put the functions in the top row, then stack each successive derivative as the row below. 2. Evaluate It — and Read the Verdict Now take the determinant. Here is the test in one line: if for even a single x_0 , the functions are linearly independent on that interval. Watch the Wronskian for collapse, term by term, to a single number: For two functions the Wronskian is the determinant W(f,g)=fg'-gf' : Since e^ 3x >0 for all x , W is never zero — so e^ x and e^ 2x are linearly independent. 3. What a Zero Does (and Doesn't) Prove The test runs one way . A nonzero Wronskian guarantees independence — but does not , on its own, prove dependence in general. Watch two pairs evaluated side by side: W=-1 , never zero — so these are linearly independent . everywhere — and indeed 2x is just , so they are dependent. You've seen how the determinant test for vectors extends to functions through the Wronskian. Scroll up to revisit any step.

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