Proof of the Wronskian test for linear independence of solutions

Builds on Wronskian Homogeneous linear equation Solution space of a homogeneous linear equation has dimension n Superposition principle Linearity of differentiation A matrix is invertible iff its determinant is a unit An injective endomorphism of a finite-dimensional space is surjective Linear independence

Proof

Let $S$ be the space of solutions on $I$ of the homogeneous linear equation. For $x \in I$ , let $\Phi_x(y) = \left(y(x), y'(x), \dots, y^{(n-1)}(x)\right)$ for $y \in S$ .
By the dimension theorem for the solution space, $\Phi_x : S \to \R^n$ is a linear bijection for each $x \in I$ .
For $x \in I$ , let $M(x)$ be the $n \times n$ matrix whose column $j$ is $\Phi_x(y_j)$ . By the definition of the Wronskian, $W(x) = \det M(x)$ .
For $c = (c_1, \dots, c_n) \in \R^n$ , let $y_c = \sum_j c_j y_j$ . By the superposition principle, $y_c \in S$ . By linearity of $\Phi_x$ , $\Phi_x(y_c) = M(x)\, c$ .
(a) implies (c). Assume (a), and suppose $W(x) = 0$ for some $x \in I$ . Then $M(x)$ is not invertible.
The linear map $c \mapsto M(x) c$ of $\R^n$ is then not bijective. By the injective endomorphism theorem, it is not injective. So there is $c \ne 0$ with $M(x) c = 0$ .
By step 4, $\Phi_x(y_c) = 0$ . Since $\Phi_x$ is injective, $y_c = 0$ on $I$ .
So $\sum_j c_j y_j = 0$ on $I$ with $c \ne 0$ . This contradicts the linear independence in (a). Hence $W(x) \ne 0$ for each $x \in I$ .
(c) implies (b). The interval $I$ is not empty, so take any $x_0 \in I$ .
(b) implies (a). Let $W(x_0) \ne 0$ , and let $c \in \R^n$ with $\sum_j c_j y_j = 0$ on $I$ . So $y_c = 0$ , and all its derivatives vanish on $I$ , by linearity of differentiation.
In particular $M(x_0)\, c = \Phi_{x_0}(y_c) = 0$ , by step 4. Since $\det M(x_0) = W(x_0) \ne 0$ , the matrix $M(x_0)$ is invertible.
Hence $c = M(x_0)^{-1} M(x_0)\, c = 0$ . So $y_1, \dots, y_n$ are linearly independent on $I$ .
The implications (a) to (c), (c) to (b) and (b) to (a) show that the three conditions are equivalent.

Idea

The columns of the Wronskian matrix at $x$ are the initial values of the solutions at $x$ . A solution is determined by its initial values at any one point. So a dependence among the columns at one point is a dependence among the solutions on all of $I$ .

Reading path

To understand this page from the ground up, read these 574 pages in order. The foundations come first. The path takes the proofs Proof that Darboux and Riemann integrals agree, Proof of the Steinitz exchange lemma by induction, Proof of uniqueness of solutions for Lipschitz vector fields, Proof of the Cauchy–Hadamard theorem, Proof of the Weierstrass M-test, Proof of the limit of a sum of sequences, Proof of Rolle's theorem, Proof of the approximation property of the supremum, Proof that every sequence has a monotone subsequence, by peaks, Proof that a subspace of full dimension is the whole space, Proof of the decomposition of a permutation into transpositions, Proof that the symmetric group is generated by transpositions, Proof of the formula for the derivative of a polynomial, Proof of the Laplace expansion along a column by transposition, Proof that open sets are the sets that are neighbourhoods of all their points, Proof of the diameter characterisation of Cauchy sequences, Proof of the completeness of the space of bounded continuous functions, Proof of the commutativity of addition by induction, Proof of the Lebesgue number lemma by a finite cover of balls, Proof of term-by-term differentiation, Proof of the epsilon-delta form of continuity, Proof that equivalence classes form a partition, Proof that the solution space of a homogeneous linear equation has dimension n, Proof of the power rule by induction, Proof that absolutely convergent series converge, Proof of the triangle inequality for norms, Proof of the sum of a finite geometric series, Proof that convergent sequences are Cauchy, Proof that a complex number times its conjugate is the square of its modulus, Proof of the Cauchy–Schwarz inequality, Proof of the Heine–Borel theorem by sequential compactness, Proof that there is a natural number n with 1/n below any positive epsilon, Proof of the chain rule with an auxiliary difference quotient, Proof of the Cauchy criterion for series, Proof of the Cauchy criterion for sequences by the Bolzano–Weierstrass theorem, Proof that the determinant is multiplicative by expanding the rows, Proof that permuting rows multiplies the determinant by the sign, Proof that the exponential series has infinite radius of convergence, Proof that disjoint cycles commute, Proof of the equality of complex numbers, Proof that continuous functions on a closed bounded interval are uniformly continuous, Proof that a matrix is invertible iff its determinant is a unit, Proof that a function with nonnegative derivative is increasing, Proof that a function is bijective iff it has a two-sided inverse, Proof that the reals form an ordered field, with the Cauchy sequence construction, Proof of the root test, Proof of continuity via closed sets, Proof that a matrix with two equal rows has determinant zero, Proof of the double negation law by a truth table, Proof that power series converge uniformly on smaller intervals inside the radius of convergence, Proof of the extreme value theorem by the Bolzano–Weierstrass theorem, Proof that compact subsets of a metric space are bounded, Proof of the Laplace expansion along a row by the Leibniz formula, Proof that compactness and sequential compactness agree in metric spaces, Proof that the exponential function is positive, Proof that subsequences of a convergent sequence converge to the same limit, Proof of the uniqueness of limits, Proof of the integral equation form of an initial value problem, Proof that the integral of a continuous function is an antiderivative, Proof that the coordinate map is a linear isomorphism, Proof of strong induction from ordinary induction, Proof that power series are differentiable inside the radius of convergence, Proof that the determinant of a transpose equals the determinant, Proof of the laws of exponents in a group, Proof of the limit of a product of sequences, Proof of the inverse in terms of the adjugate, Proof that uniform convergence of derivatives allows term-by-term differentiation, Proof that the absolute value is multiplicative, Proof of the squeeze theorem for sequences, Proof of the existence of square roots of nonnegative reals, Proof of Pascal's rule from the recursive definition, Proof of the linearity of differentiation, Proof of the mean value theorem for integrals, Proof that a continuous image of a compact set is compact, Proof that an injective endomorphism of a finite-dimensional space is surjective, Proof that compact subsets of a Hausdorff space are closed, Proof of epsilon-delta continuity at a point in metric spaces, Proof of Riemann's criterion for integrability, Proof of the sequential characterization of limits, Proof that Cauchy sequences are bounded, Proof of the description of the open sets of a metric space, Proof of the Banach fixed point theorem by iteration, Proof of the triangle inequality for real numbers, Proof that an intersection of subgroups is a subgroup, Proof of the epsilon-N definition of the limit of a sequence, Proof of the first fundamental theorem of calculus, Proof that compactness of a subset can be tested with open sets of the ambient space, Proof that the sign is a homomorphism, Proof of the comparison test, Proof of the existence of the infimum by negation, Proof of the equivalence of norms in finite dimension by compactness of the unit sphere, Proof of the determinant of the inverse, Proof that every spanning set contains a basis, by Zorn's lemma, Proof that the determinant of the identity matrix is one, Proof of Fermat's theorem on stationary points, Proof that limits preserve non-strict inequalities, Proof that a linear map is injective iff its kernel is trivial, Proof that every linearly independent set extends to a basis, by Zorn's lemma, Proof that closed sets contain their limits, Proof of the linearity of the integral, Proof of the factorial formula for binomial coefficients by induction, Proof of Zorn's lemma, Proof that metric spaces are Hausdorff, Proof that an iterate of the Picard operator is a contraction, Proof of the second fundamental theorem of calculus, Proof of global existence for globally Lipschitz vector fields by steps of uniform length, Proof of the intermediate value theorem by the supremum, Proof of the binomial theorem by induction, Proof that exponentials dominate powers in sequences, by the ratio of consecutive terms, Proof that differentiable functions are continuous, Proof of the decomposition of a permutation into disjoint cycles, Proof of the reduction of a higher-order equation to a first-order system, Proof of the product of a matrix and its adjugate by Laplace expansion, Proof of the epsilon-delta form of the limit of a function, Proof of the well-ordering principle from strong induction, Proof that n independent or n spanning vectors in an n-dimensional space form a basis, Proof of the Bolzano–Weierstrass theorem by a monotone subsequence, Proof of the division algorithm, Proof of Grönwall's inequality, Proof of the Archimedean property in an Archimedean ordered field, Proof of the sign of a cycle, Proof that finite-dimensional normed spaces are complete, Proof that a continuous real function on a compact set attains its bounds, Proof of the Leibniz formula from the defining properties of the determinant, Proof of continuous dependence on initial data by Grönwall's inequality, Proof of the uniqueness of inverses, Proof that a composition of continuous functions is continuous, Proof of the exponential of a sum by the Cauchy product, Proof of the additivity of the integral over intervals, Proof that a decreasing sequence bounded below converges, Proof of the term test, Proof that the exponential function is its own derivative, by term-by-term differentiation, Proof that the reals are Archimedean from the completeness axiom, Proof of the ratio test, Proof that convergent sequences are bounded, Proof of the product rule, Proof of the monotone convergence theorem for sequences, Proof that continuity is equivalent to sequential continuity, Proof that a subspace of a finite-dimensional space has smaller or equal dimension, Proof that the determinant is multilinear in the rows, Proof that open balls are open, Proof of the monotonicity of the integral, Proof of the boundedness theorem by Bolzano–Weierstrass, Proof of the ratio formula for the radius of convergence, Proof of the superposition principle, Proof that a set is open if and only if its complement is closed, Proof of the rank–nullity theorem by extending a basis of the kernel, Proof of the fundamental theorem of calculus, Proof that a series of nonnegative terms converges iff its partial sums are bounded, Proof of the subgroup criterion, Proof of the Picard–Lindelöf theorem, Proof of the mean value theorem from Rolle's theorem, Proof that the Cauchy product of absolutely convergent series converges absolutely to the product, Proof of continuity via open sets in metric spaces, Proof that continuous functions are Riemann integrable, Proof of the constant multiple rule.

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Proof of the Wronskian test for linear independence of solutions proof

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Wronskian test for linear independence of solutions

Tag	`5J73GZAZKS`
URI	`m2://5J73GZAZKS`
Kind	proof
Writing	W0 stub
MSC	34A30
Proves	Wronskian test for linear independence of solutions
Uses	Wronskian, Homogeneous linear equation, Solution space of a homogeneous linear equation has dimension n, Superposition principle, Linearity of differentiation, A matrix is invertible iff its determinant is a unit, An injective endomorphism of a finite-dimensional space is surjective, Linear independence
Changed	4 November 2026

Proof of the Wronskian test for linear independence of solutions

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