A projection is a linear operator $P$ with $P^2=P$. It acts as the identity on its image and as zero on its kernel. It is orthogonal when image and kernel are orthogonal, equivalently $P=P^*$ in finite-dimensional inner-product coordinates. Idempotent operators need not be orthogonal projections.
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- Definition of (orthogonal) projection of one vector onto another vectorFor $u\ne0$, the orthogonal projection of $v$ onto the line spanned by $u$ is the unique vector $cu$ such that...
- Formula for the (orthogonal) projection of one vector onto another vectorThe projection formula is $\operatorname{proj} u(v)=u\,\langle u,v\rangle/\ u\ ^2$ for $u\ne0$. The projection matrix...
- Definition of (not necessarily orthogonal) projection onto a component of a direct sumIf $V=U\oplus W$, define the projection onto $U$ along $W$ by $P(u+w)=u$. Unique decomposition makes this...
- Definition of (orthogonal) projection onto a subspaceFor a subspace $W$ of a finite-dimensional inner-product space, $P Wv$ is the unique element of $W$ whose residual...
- The projection of a vector which is in a subspace is the vector itself.If $v\in W$, then $P Wv=v$, because $v=v+0$ is its orthogonal decomposition. Therefore every nonzero vector in the...
- The (orthogonal) projection of a vector onto a subspace is the point in the subspace closest to the vector.For $w=P Wv$ and any $u\in W$, orthogonality gives $\ v-u\ ^2=\ v-w\ ^2+\ w-u\ ^2$. The second term is nonnegative...
- Formula for the coordinates of the projection of a vector onto a subspaceIf $u 1,\ldots,u k$ is an orthogonal basis of $W$, then $P Wv=\sum i u i\langle u i,v\rangle/\ u i\ ^2$. These...
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