Unterräume

Unterräume are a fundamental concept in linear algebra. A subset W of a vector space V is called a subspace if it satisfies three conditions: it contains the zero vector, it is closed under vector addition, and it is closed under scalar multiplication. This means that if you take any two vectors from W and add them together, the resulting vector must also be in W. Similarly, if you take any vector from W and multiply it by any scalar, the result must also be in W.

These conditions ensure that a subspace itself forms a vector space under the same operations as the larger space V. For example, the set of all vectors in R3 that lie on a plane passing through the origin is a subspace of R3. The zero vector (0,0,0) is on the plane. If you add two vectors on the plane, the sum remains on the plane. If you scale a vector on the plane, it stays on the plane.

Important examples of subspaces include the trivial subspace, which contains only the zero vector, and the entire vector space itself. Any line passing through the origin in R2 or R3 is also a subspace. The intersection of any collection of subspaces of a vector space is also a subspace. However, the union of two subspaces is generally not a subspace unless one is contained within the other. Understanding subspaces is crucial for many areas of mathematics and its applications, including solving systems of linear equations and analyzing linear transformations.