Vector space/Projection/Introduction/Section
For a direct sum decomposition of a -vector space , the linear mapping
is called the first projection (or projection onto with respect to the given decomposition or projection onto along ) and, accordingly,
the second projection of this decomposition. Since and are linear subspaces of , it is reasonable to call the composed mapping
a projection as well. Then we have a projection in the sense of the following definition.
Let be a field, a -vector space and a linear subspace. A linear mapping
is called a projection of onto , if and
holds.Let be a finite-dimensional -vector space and , , a basis of . For a subset , set
the linear subspace corresponding to . Moreover, let
the corresponding projection. The image of this projection is . On , this mapping is the identity, one can also consider this mapping as
The kernel of this mapping is
For , equipped with the standard basis, we consider subsets with two elements and the corresponding projections (in the sense of example) onto the coordinate planes. The projections are
The images of some object in under these projections carry names like ground view etc.
For a subset with one element, the projections map to an axis.
A more abstract definition is the following that does not refer to a linear subspace.
The identity and the zero mapping are projections.
Let be a field and a -vector space. For a direct sum decomposition
the projection onto is a projection in the sense of Definition. Such a projection
gives a decomposition
Let be the projection onto . Write with . Then we have
and hence
Suppose now that
is an endomorphism with
Let . Then there exists some such that
Then
This means that the intersection of these linear subspaces is the zero space. For an arbitrary , we write
Here, the first summand belongs to the image and, because of
the second summand belongs to the kernel. Therefore, we have a direct sum decomposition.