Linear variety

linear manifold, affine subspace, flat

A subset $M$ of a (linear) vector space $E$ that is a translate of a linear subspace $L$ of $E$, that is, a set $M$ of the form $x_0 + L$ for some $x_0$. The set $M$ determines $L$ uniquely, whereas $x_0$ is defined only modulo $L$: $$ x_0 + L = x_1 + N $$ if and only if $L = N$ and $x_1 - x_0 \in L$. The dimension of $M$ is the dimension of $L$. A linear variety corresponding to a subspace of codimension 1 is called a hyperplane.

A linear variety may be alternatively characterised as a non-empty subset $M$ of $E$ such that the set $L =\{ m_1 - m_2 : m_1,m_2 \in M \}$ is a linear subspace of $E$; or such that for a fixed $m_0 \in M$ the set $L =\{ m_1 - m_0 : m_1 \in M \}$ is a linear subspace; or as a set closed under linear combinations $\sum_i \lambda_i m_i$ where $m_i \in M$ and $\sum_i \lambda_i = 1$.

The intersection of any family of linear varieties is again a linear variety.

References

• N. Bourbaki, "Algebra I: Chapters 1-3", Elements of mathematics, Springer (1998) ISBN 3-540-64243-9