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Section4.2Projections Onto Subspaces

SubsectionThe Assignment

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

SubsectionLearning 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.

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 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.

SubsectionSage 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.

SubsectionExercises

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 \(x-2y+3z = 0\) 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 3-dimensional hyperplanes meet.)

Find the orthogonal projection of the vector \(\left(-3,1,-3,1\right)\) on this plane.