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linear-algebra-book-changes.txt
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linear-algebra-book-changes.txt
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Applications
------------
[ ] Write an application for traffic analysis (grid of streets, etc.)
[ ] Write an application of ray-based tomography:
http://www.allacademic.com/meta/p_mla_apa_research_citation/4/3/5/8/1/p435815_index.html,
http://en.wikipedia.org/wiki/X-ray_computed_tomography#Process
[ ] When watching football, you see a timer projected onto the field. The
nice thing about a football field is that there are straight lines
that give us a way to calculate the linear transformation. So give
the students a picture of the field from an angle, and have them
calculate the linear transformation needed to project some text onto
the field and have it appear as part of the field. Maybe even have
them actually have them measure the angle and distances.
Somewhere
---------
[ ] Block matrices (in the first chapter? Maybe the project for the
first chapter?)
[ ] Comparison of the sizes of linearly independent and spanning sets.
For now, I put a few statements in chapter 5 about this, but it would
be good to see it earlier.
[ ] More emphasis in chapter 2 that vector spaces are spans of vectors
[ ] More emphasis in chapter 3 that the properties of vector space
operations are the patterns that we take from addition and
multiplication of vectors in R^n and M_{mn} and P_n
[ ] Move Inner Products before Changing Bases, and maybe even before
Linear Transformations. The idea is to introduce what nice bases look
like before talking about transforming bases, so people have an idea
why they might want to change bases. Actually, having inner products
right after vector spaces might make a lot of sense because we are
talking about projecting vectors, orthogonalization, etc.
Patterns Chapter
----------------
[ ] Change order of vector space chapter to have
* Change what we mean by "vector", and have subspaces as spans of
vectors. Emphasize bases and coordinate vectors
* Change what we mean by addition/multiplication, and view the rules
as an extension of the properties in R^n, P_n, and M_n properties.
* Talk about logarithmic space as an example of the common
properties.
[ ] Add HW: Having another problem or two in
chapter 3 asking them to find det(AB), det(A-1), det(A2), det(3A)
given that det(A)=4 and det(B)=5 and that A is a 3x3 matrix would help
them look up and see and use the theorems a bit more.
Linear Transformations Chapter
------------------------------
[ ] LU decomposition project (especially after the elementary matrices
[ ] If inner products are before the linear transformation chapter, then
householder reflections and QR decomposition might make a great
project (as it deals with inner products and with geometry)
[ ] Add knowing |AB|=|A||B| in with the objectives
[ ] Add more HW dealing with the determinant property |AB|=|A||B|
Changing Bases chapter
----------------------
[ ] Add (AB)^{-1}=B^{-1}A^{-1} to the objectives
[ ] Add HW problem that says something like to change from B1 to B2, we
need S. To change from B2 to B3, we need T. What matrix gets us
from B3 to B1?
[x] Switch order of subsections in 4.2
--------------------------------------
In keeping with the philosophy that students get lost when lots of definitions and terminology come before concrete examples, I propose reversing the subsections in 4.2 (so it would be "Standard matrix transforms", "Linear Transformations", "General vocabulary of functions"; see my revisions at https://bitbucket.org/jasongrout/linear-algebra/downloads to check the section numbers). This does several things:
1. It puts finding the matrix of a linear transformation just after we've talked about what a matrix does in a linear transformation. This is very natural (i.e., first is given a matrix, what is the geometry, then given the geometry, what is the matrix?) and very easy to do.
2. Then it is very easy to talk about what a linear transformation is. For example, after I had the class find a bunch of matrices given the geometry of the transformation, I talked about how f(2,1)=2f(1,0)+f(0,1), and the fact that the linear transformation totally depended on just the values of those two vectors. This was very natural since they saw obviously that the results of those two vectors gave the entire matrix for a 2 by 2 matrix. That led naturally into a discussion of linear transformations, the definition, and examples.
3. *then* we formalize the function vocabulary in preparation for talking about the images and kernel of a transformation. That way the vocabulary comes right before we need it, instead of at the beginning of the chapter, not to be used until the next section. It also comes after the motivation of linear transformations, inverse matrices, etc.