Math 2500: Linear Algebra

Subsection 4.2.1 The Assignment

• Read Chapter 4 section 2 of Strang.
• Read the following and complete the exercises below.

Subsection 4.2.2 Learning Goals

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.

Subsection 4.2.3 Discussion: 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 ground--its 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.

Subsection 4.2.4 Sage and Orthogonal Projection

Sage has no built-in 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.

Subsection 4.2.5 Exercises