Teilraum

Teilraum, also known as a subspace in English, is a fundamental concept in linear algebra. A Teilraum of a vector space V is a subset W of V that is itself a vector space under the same operations of vector addition and scalar multiplication as defined on V.

For a subset W of a vector space V to be a Teilraum, it must satisfy three conditions. First, the zero vector of V must be an element of W. Second, W must be closed under vector addition, meaning that for any two vectors u and v in W, their sum u + v must also be in W. Third, W must be closed under scalar multiplication, meaning that for any vector w in W and any scalar c, the product c * w must also be in W.

If these three conditions are met, then W is a Teilraum of V. Examples of Teileräume include the set containing only the zero vector, the entire vector space V itself, and lines or planes passing through the origin in R^n. The concept of a Teilraum is crucial for understanding linear transformations, eigenvalues, and eigenvectors. It allows mathematicians to study the structure of vector spaces by focusing on their substructures that retain the properties of a vector space.