Linear independence of vector functions

For a system of ODEs, linear independence of vector-valued solutions is the analog of linear independence for scalar functions. $m$ vector functions ${\mathbf{v}}_1(t),\dots ,{\mathbf{v}}_m(t)$ are linearly dependent on an interval $I$ if there exist constants $c_1,\dots ,c_m$ (not all zero) such that

$c_1{\mathbf{v}}_1(t)+\cdots +c_m{\mathbf{v}}_m(t)=0\text{for all }t\in I$

Otherwise, they’re linearly independent.

Test via determinant

For $n$ vector functions in ${\mathbb{R}}^n$ (so a square matrix when arranged as columns), there’s a useful sufficient condition.

Proposition: if there exists $t_0\in I$ such that

$\det \left[\begin{array}{cccc}{v}_{1,1}(t_0)& v_{2,1}(t_0)& \cdots & v_{n,1}(t_0)\\ ⋮& ⋮& & ⋮\\ v_{1,n}(t_0)& v_{2,n}(t_0)& \cdots & v_{n,n}(t_0)\end{array}\right]\ne 0$

then ${\mathbf{v}}_1,\dots ,{\mathbf{v}}_n$ are linearly independent on $I$.

This determinant is the Wronskian for vector functions — the matrix analog of the scalar Wronskian.

The converse is not true: the Wronskian can be zero everywhere without the functions being linearly dependent (rare, requires careful construction).

For solutions of linear ODE systems

For solutions of a linear homogeneous system ${\mathbf{x}}^{\prime }=A(t)\mathbf{x}$, the situation is cleaner. The following are equivalent:

1. ${\mathbf{v}}_1,\dots ,{\mathbf{v}}_n$ are linearly independent on $I$.
2. The Wronskian $W[{\mathbf{v}}_1,\dots ,{\mathbf{v}}_n](t)\ne 0$ for every $t\in I$.
3. There exists $t_0\in I$ such that $W[{\mathbf{v}}_1,\dots ,{\mathbf{v}}_n](t_0)\ne 0$.

In words: for solutions of a linear homogeneous ODE system, the Wronskian is either identically zero or never zero on $I$. There’s no in-between.

This is the system-level analog of Abel’s theorem for scalar ODEs.

General solution

If ${\mathbf{x}}_1(t),\dots ,{\mathbf{x}}_n(t)$ are $n$ linearly independent solutions of the homogeneous system ${\mathbf{x}}^{\prime }=A(t)\mathbf{x}$ on $I$, then every other solution has the form

$\mathbf{x}(t)=c_1{\mathbf{x}}_1(t)+c_2{\mathbf{x}}_2(t)+\cdots +c_n{\mathbf{x}}_n(t)$

for some constants $c_1,\dots ,c_n$. This is the general solution, and the constants are determined by initial conditions.

The set $\{{\mathbf{x}}_1,\dots ,{\mathbf{x}}_n\}$ is called a fundamental set of solutions, and the matrix $\Psi (t)=[{\mathbf{x}}_1(t)|{\mathbf{x}}_2(t)|\cdots |{\mathbf{x}}_n(t)]$ is a fundamental matrix.

Why $n$ solutions

For an $n$-dimensional system, the solution space is $n$-dimensional. So $n$ linearly independent solutions span the entire space, and you can express any solution as their linear combination.

This is the Representation theorem applied to systems.

Why this matters in practice

When you find solutions of a homogeneous system via eigenvalues — getting one solution per eigenvalue/eigenvector pair — you need to verify they’re linearly independent before declaring the general solution.

For a $2\times 2$ system with two distinct real eigenvalues, the two eigenvectors are automatically linearly independent (eigenvectors corresponding to distinct eigenvalues are always independent). So ${\mathbf{v}}_1{e}^{{\lambda }_{1}t}$ and ${\mathbf{v}}_2{e}^{{\lambda }_{2}t}$ are linearly independent solutions, and any solution is a combination.

For repeated eigenvalues with insufficient eigenvectors, you have to construct a generalized eigenvector to get the second linearly independent solution.

For the underlying theorem, see Representation theorem. For systems-of-ODEs context, see System of first-order linear ODEs.