 Read Chapter 4 section 2 of Strang.
 Read the following and complete the exercises below.
Before class, a student should be able to:
 Compute the projection of a vector onto a line.
 Find the projection matrix which computes the projections of vectors onto a given line.
 Draw the schematic picture of a projection: the line, the vector, the projected vector, and the difference.
Sometime after class, a student should be able to:
 Compute the projection of a vector onto a subspace.
 Find the projection matrix which computes the projections of vectors onto a given subspace.

Explain the process for finding the equations which determine the projection matrix, and say
why the transpose makes an appearance.
SubsectionDiscussion: Orthogonal Projections
One good use of the geometry in \(\mathbb{R}^n\) is the concept of
orthogonal projection. The basic idea is to mimic the behavior of
shadows under sunlight. Our everyday experience leads us to thinking
about the projection of a vector onto a plane (the groundits roughly
a plane), but if you imagine holding out a pencil you can summon up the
visual of projection onto a line, too.
The key concept is to use the basic condition of orthogonality
(\(u \cdot v = 0\)) to figure things out.
Note that everything in this section is done by projecting onto
subspaces! This is a bit of a restriction. In practice, this
restriction can be removed by translating your whole problem to have a
new origin.
SubsectionSage and Orthogonal Projection
Sage has no builtin commands for orthogonal projections. But let us recall
those parts of Sage that will be useful right now:
Sorry, that matrix isn't even square, so it can't be invertible. But
this will be:
Finally, this makes sense:
This process should have some basic properties. Let's check them.
So \(A^TA\) is square, symmetric, and invertible.
Also as expected.
Task112
Find the projection matrix which computes projections of vectors in
\(\mathbb{R}^2\) onto the line \(3x+2y=0\).
(Since it goes through zero, it is a subspace.)
Find the orthogonal projection of the vector
\(\left( 17,3 \right)\) onto this line.
Task113
Find the projection matrix which computes projections of vectors in
\(\mathbb{R}^3\) onto the line which is the intersection of the
planes \(x2y+3z = 0\) and \(y+2z=0\). (Again, that is a subspace.)
Find the orthogonal projection of the vector \(\left(1,1,1\right)\)
onto this line.
Task114
Find the projection matrix which computes projections of vectors in
\(\mathbb{R}^3\) onto the plane \(2x + y +3z = 0\).
Find the orthogonal projection of the vector \(\left( 9,7,5\right)\)
onto this plane.
Task115
Find the projection matrix which computes projections of vectors in
\(\mathbb{R}^4\) onto the plane which is the intersectoin of
\(5x+y +w=0\) and \(z+y+z+w=0\). (This subspace is the 2
dimensional plane where these two 3dimensional hyperplanes meet.)
Find the orthogonal projection of the vector
\(\left(3,1,3,1\right)\) on this plane.