The concept of orthogonality is dependent on the choice of inner product. So assume first that we are working with the standard dot product in . We say two vectors , are orthogonal if they are non-zero and ; we indicate this by writing . Orthogonality with respect to this standard inner product corresponds to our usual notion of perpendicular (as we shall see below). More generally, a collection of non-zero vectors is said to be orthogonal if they are pairwise orthogonal; in other words, for all .
The notion of orthogonality extends to subspaces. Thus if are two non-zero subspaces, we say and are orthogonal () if . As with a collection of vectors, a collection of subspaces is orthogonal iff it is pairwise orthogonal: .
Let . Then it is easy to check that and are orthogonal (geometrically, they are
represented by two lines in passing through the origin and forming a angle between
them).
If is a subspace, its orthogonal complement is given by . is the largest subspace of for which every non-zero vector in the subspace is orthogonal to every non-zero vector in .
Orthogonality is connected to the property of linear independence.
- Proof
- Suppose there exist scalars with . Then for each one has which implies as .