= Fundamental system of solutions
{title2=$y_1,\ldots,y_n$}
A fundamental system for a homogeneous <linear ordinary differential equation> of order $n$ on an interval is a <basis> of its <vector space> of <homogeneous solutions>. Every solution has a unique expression $\sum_{j=1}^nc_jy_j$ with constant <coefficients>. Under the usual <continuity> assumptions and a nonvanishing leading <coefficient>, a nonzero <Wronskian> at one point proves that $n$ solutions form such a system: their initial-value <vectors> are independent and span all possible initial data, and the <Picard-Lindelöf theorem> supplies uniqueness. For a first-order <vector> system, the corresponding columns form a <fundamental matrix>.
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