Editing Projection (linear algebra) (section)

===Orthogonal projection===
For example, the function which maps the point <math>(x,y,z)</math> in three-dimensional space <math>\mathbb{R}^3</math> to the point <math>(x,y,0)</math> is an orthogonal projection onto the ''xy''-plane. This function is represented by the matrix
<math display="block">P = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 0 \end{bmatrix}.</math>

The action of this matrix on an arbitrary [[Euclidean vector|vector]] is
<math display="block">P \begin{bmatrix} x \\ y \\ z \end{bmatrix} = \begin{bmatrix} x \\ y \\ 0 \end{bmatrix}.</math>

To see that <math>P</math> is indeed a projection, i.e., <math>P = P^2</math>, we compute
<math display="block">P^2 \begin{bmatrix} x \\ y \\ z \end{bmatrix} = P \begin{bmatrix} x \\ y \\ 0 \end{bmatrix} = \begin{bmatrix} x \\ y \\ 0 \end{bmatrix} = P\begin{bmatrix} x \\ y \\ z \end{bmatrix}.</math>

Observing that <math>P^{\mathrm T} = P</math> shows that the projection is an orthogonal projection.