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A linear transformation $P:V\rightarrow V$ of a vector space $V$ is called a projection if it acts like the identity on its image. This condition can be more succinctly expressed by the equation \begin{equation} \label{eq:proj} P^2 = P. \end{equation}
Proposition 1 If $P:V\rightarrow V$ is a projection, then its image and the kernel are complementary subspaces, namely \begin{equation} \label{eq:comp} V = \ker P \oplus \img P. \end{equation}
Proof. Suppose that $P$ is a projection. Let $v\in V$ be given, and set $$u=v-Pv.$$ The projection condition (![[*]](http://images.planetmath.org:8080/cache/objects/3208/js//usr/share/latex2html/icons/crossref.png "[*]") ) then implies that $u\in \ker P$ and we can write $v$ as the sum of an image and kernel vectors: $$v = u + Pv.$$ This decomposition is unique, because the intersection of the image and the kernel is the trivial subspace. Indeed, suppose that $v\in V$ is in both the image and the kernel of $P$ Then, $Pv=v$ and $Pv=0$ and hence $v=0$ QED
Conversely, every direct sum decomposition $$V = V_1 \oplus V_2$$ corresponds to a projection $P:V\rightarrow V$ defined by $$ Pv =\begin{cases} v & v\in V_1 \\ 0 & v\in V_2 \end{cases}$$
Specializing somewhat, suppose that the ground field is $\reals$ or $\cnums$ and that $V$ is equipped with a positive-definite inner product. In this setting we call an endomorphism $P:V\rightarrow V$ an orthogonal projection if it is self-dual $$P\adj = P,$$ in addition to satisfying the projection condition ().
Proposition 2 The kernel and image of an orthogonal projection are orthogonal subspaces.
Proof. Let $u\in\ker P$ and $v\in \img P$ be given. Since $P$ is self-dual we have $$0 = \langle Pu,v\rangle = \langle u,Pv\rangle = \langle u,v\rangle.$$ QED
Thus we see that a orthogonal projection $P$ projects a $v \in V$ onto $Pv$ in an orthogonal fashion, i.e. $$\langle v-Pv,u\rangle = 0$$ for all $u\in \img P$
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"projection" is owned by rmilson.
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orthogonal projection |
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Cross-references: onto, projects, orthogonal, addition, self-dual, endomorphism, inner product, ground field, direct sum, conversely, QED, subspace, intersection, decomposition, vectors, sum, implies, proof, complementary subspaces, kernel, equation, image, identity, vector space, linear transformation
There are 75 references to this entry.
This is version 5 of projection, born on 2002-07-26, modified 2004-02-25.
Object id is 3208, canonical name is Projection.
Accessed 19608 times total.
Classification:
| AMS MSC: | 15A21 (Linear and multilinear algebra; matrix theory :: Canonical forms, reductions, classification) | | | 15A57 (Linear and multilinear algebra; matrix theory :: Other types of matrices ) |
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Pending Errata and Addenda
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