Vertikalvektor

Vertikalvektor is a term used in mathematics, specifically within the context of vector spaces and linear algebra. It refers to a vector that is orthogonal (perpendicular) to a given subspace. In simpler terms, if you have a plane or a line within a larger space, a vertikalvektor would be a vector that points straight out from that plane or line, at a 90-degree angle to every direction within that plane or line.

The concept of a vertikalvektor is closely related to orthogonal complements. The orthogonal complement of a subspace W, denoted as W^⊥, is the set of all vectors that are orthogonal to every vector in W. The vertikalvektor belongs to this orthogonal complement. This idea is fundamental in areas like linear regression, where finding the closest point in a subspace to a given vector often involves projecting that vector onto the subspace, and the difference between the original vector and its projection is a vertikalvektor.

The existence and properties of vertikalvektors depend on the underlying vector space and the inner product defined on it. In Euclidean space with the standard dot product, orthogonality has its usual geometric interpretation. The notion of a vertikalvektor is essential for understanding concepts like orthogonal projections, least squares solutions, and the structure of linear transformations.