In linear algebra, a Euclidean subspace is a set of vectors that is closed under addition and scalar multiplication. Geometrically, a subspace is a flat in n-dimensional Euclidean space that passes through the origin. Examples of subspaces include the solution set to a homogeneous system of linear equations, the subset of Euclidean space described by a system of homogeneous linear parametric equations, the span of a collection of vectors, and the null space, column space, and row space of a matrix. In abstract linear algebra, Euclidean subspaces are important examples of vector spaces. In this context, a Euclidean subspace is simply a linear subspace of a Euclidean space.

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