# How To Find The Matrix Of An Orthogonal Projection

**To nd the matrix of the orthogonal projection onto V the way we rst discussed takes three steps.**

**How to find the matrix of an orthogonal projection**.
Its rows are mutually orthogonal vectors with unit norm so that the rows constitute an orthonormal basis of V.
P A inv AA A A inv AA A A inv AA A A inv A A A P where several properties of inverses and transposes have been used that were demontrated several videos ago.
Use Orthogonal Projection Matrix to find the matrix A of the orthogonal projection onto W s p a n 1 1 1 1 1 9 5 3.

First we apply Gram-Schmidt Process to W s p a n. I realize that this is not possible with an orthographic projection matrix and that I will have to set up a new perspective projection matrix. Let W be the subspace of R2 spanned by 1 1.

R n R n by T x x W and let B be the standard matrix for T. In fact a simple identity matrix with a slight modification will do the trick. You might think that orthographic projections are of no use today.

So instead of dividing by aT a we now have to multiply by AT A1 In n dimensions ˆ AT A1 AT b. Figuring out the transformation matrix for a projection onto a subspace by figuring out the matrix for the projection onto the subspaces orthogonal compleme. 2 Turn the basis v i into an orthonormal basis u i using the Gram-Schmidt algorithm.

If we view the vector v1 as an n 1 matrix and the scalar v1 x as a 1 1 we can write. We can rewrite the equation AT b Axˆ 0 as. Projection onto a subspace.

Then compute v q which will be the desired projection. Finding a standard matrix for a linear transformation that is the orthogonal projection of a vector onto the subspace 3x4z0. The orthographic projection also sometimes called oblique projection is simpler than the other type of projections and learning about it is a good way of apprehending how the perspective projection matrix works.