qbobwatson/gt-linalg

Interactive Linear Algebra, free online textbook at Georgia Tech

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gt-linalg's Issues

Feedback on solution-sets

In reference to version cebba0d48f43784542dd70c9421777691a9419c6

The proof of the key observation only shows one direction.

Feedback on solution-sets

In reference to version cebba0d48f43784542dd70c9421777691a9419c6

Isn't the second blue box something they should have seen already in 2.3?

Feedback on row-reduction

In reference to version 79e5138

I think at the beginning we should show systems of equations whose corresponding matrices are in REF or RREF, and pointing out that in the first case you can easily find the answer by back substitution and in the second case you can just see the solution. This motivates the whole thing. I think we both have something like this in our slides.

Feedback on systems-of-eqns

In reference to version f6ae88a

In class I noticed that many students did not know how to locate a point in R^3. Here is an idea for a demo. The demo spits out a random integer point in R^3 and shows a dot there. Then you play with the x, y, and z slider bars in order to determine the coordinates of the point.

Feedback on parametric-form

In reference to version 79e5138

The first example in the free variables section is explaining new material. It should not be in a knowl.

Feedback on row-reduction

In reference to version 79e5138

What does the word profit mean here?

Feedback on solution-sets

In reference to version d2646db

There is an example in a knowl (solution is a line) and then the discussion of the example continues outside the knowl.

Feedback on solution-sets

In reference to version cebba0d48f43784542dd70c9421777691a9419c6

For the last blue box, we could add: in one case you are asking which x's work for a given b, and in the other case you are asking which b's work for a given x.

Feedback on parametric-form

In reference to version 79e5138

Why is "consistent" in bold in the definition of a free variable?

Feedback on spans

In reference to version a327fdb

We have:

We saw in the above example that the system of equations (3.2.2) is consistent. Equivalently, this means that the vector equation (3.2.1) has a solution.

I think this is hard to read. I would repeat the system of equations and the vector equation. What do you think?

Feedback on row-reduction

In reference to version cebba0d48f43784542dd70c9421777691a9419c6

typo: subjcet

Feedback on vectors

In reference to version 30f811f

Maybe "scale" instead of "scalar multiply"?

Feedback on spans

In reference to version a327fdb

I would have all three interactives at the bottom. It can be confusing sometimes when there is an interactive between two figures.

Feedback on solution-sets

In reference to version cebba0d48f43784542dd70c9421777691a9419c6

In the section "Homogeneous Systems" there are three ideas: homogeneous systems, geometric descriptions of solutions to homogeneous systems, and parametric vector form for h. systems. I feel like these should be separated.

Feedback on solution-sets

In reference to version cebba0d48f43784542dd70c9421777691a9419c6

The discussion of dimension should come earlier. There's already a bit in the homogeneous case where one free variable is a line, two free variables is a plane, etc. I think this should be a blue box:

If a system of linear equations in n variables has k free variables, then the solution set is a k-dimensional plane in R^n

And after the blue box: A 0-dimensional plane is a point, a 1-dimensional plane is a line, and a 2-dimensional plane is what we usually call just a plane. After that we don't have special names for 3-dimensional planes, 4-dimensional planes, etc.

Actually, we don't have an official definition of "dimension" yet (this comes in Section X.Y). Intuitively, the dimension of a solution set is the number of parameters you need to describe where you are in the solution set. For example, it only takes one parameter to say where you are on a line, even if the line is in R^3 or R^{100}. Similarly, if I know you are on Peachtree Street, you only need to give me one parameter (the street address) to tell me where you are.

Feedback on solution-sets

In reference to version d2646db

Parametric vector form is introduced in a knowled example before it is introduced for real. Isn't that awkward?

Feedback on systems-of-eqns

In reference to version 79e5138

As discussed over email, let's search for "element" throughout the text and replace it with "point" when appropriate.

Feedback on solution-sets

In reference to version cebba0d48f43784542dd70c9421777691a9419c6

there is an extraneous comma in the first observation

Feedback on systems-of-eqns

In reference to version 79e5138

In my slides (and I think in yours as well) we show that students that row replacement pivots one plane around the solution set. Is it worth putting this in the book? It could be a remark.

(Does this really explain the term "pivot"? It at least relates the word pivot to the geometric picture.)

Feedback on solution-sets

In reference to version cebba0d48f43784542dd70c9421777691a9419c6

I don't understand the remark after the Key Obs

Feedback on row-reduction

In reference to version f6ae88a

#2 in the definition of REF is clumsy. The second time "leading entry" is used, it is not specified that it is the first nonzero entry. Also, I don't think it is explained what "leading" means ever.

Feedback on solution-sets

In reference to version d2646db

In the first inhomogeneous example where the solution set is a plane. It says "compare this example". It would be much better to say: Earlier we solved the associated homogeneous matrix equation Ax=0....

Feedback on parametric-form

In reference to version d2646db

Hi!
On the last matrix from the section 2.3 Parametric Form of your textbook I think that there may be a number 1 missing in the third column.

Sorry if I'm wrong and thanks for the awesome material!

Miriam Koga

Feedback on matrix-equations

In reference to version cebba0d48f43784542dd70c9421777691a9419c6

Let's change "easy to verify". I don't think this is easy for the typical student.

Feedback on solution-sets

In reference to version cebba0d48f43784542dd70c9421777691a9419c6

The sentence right after the Key Obs seems to be part of the Key Obs, maybe an "in particular".

Also, do we talk about translating planes anywhere? For some students this is intuitive, but for most students it is not.

Feedback on parametric-form

In reference to version a327fdb

Where it says:

We have found all solutions

What about having the curly brace and then

x = whatever
y = whatever
z = any real number

This makes it easier to go to the parametric form

(whatever,whatever,z)

Feedback on solution-sets

In reference to version cebba0d48f43784542dd70c9421777691a9419c6

In the second observation there is a link to an example that is not really needed. I think this is undesirable. Why make the student click on something, where they then have to figure out where the example is coming from, and read the example, when the whole thing is easily explained without the link?

Don't mean to make a mountain out of this one molehill, but I think it would be better to reduce the number of hyperlinks in general.

Feedback on systems-of-eqns

In reference to version 79e5138

You convinced me that it is a good idea to hint towards the parametric form in the first section. An issue I see is that we are doing two things at once: pointing out that solutions to single equations are planes, and also saying that planes have two descriptions. Why not get the first point across first, and then the second?

So after Pictures of Solution Sets there could be a final subsection called Parametric Description of Solution Sets, where we go through the same examples and give the alternate descriptions. I would give a couple more examples of planes, such as the plane z=1 and x+y=1 (to compare with the 2D example).

Another troubling thing is that "implicit" and "parametric" are not opposites. This should be addressed. Should we think of the parametric form as the explicit form? In what way is it more explicit? I guess you are giving explicit instructions for how to list the points in the plane, right?

Feedback on parametric-form

In reference to version 79e5138

Should part of the recipe for parametric form be how to write the answer as an n-tuple? I feel like this should be an appendix to the recipe. Also it would be good after the recipe to show a sample parametric form and then the corresponding n-tuple.

Make Overview un-numbered

Chapter 2 should be called chapter 1

Feedback on vectors

In reference to version a327fdb

The first interactive for linear combinations doesn't say "interactive"

Feedback on solution-sets

In reference to version cebba0d48f43784542dd70c9421777691a9419c6

At the very end of the first recipe, did you want the word "as" in the last line? I'm not sure.

Feedback on vectors

In reference to version 30f811f

I feel like we should add to the end of this section a sample question of the form: is this vector a linear combination of these other ones? Or maybe move that chunk from the next section to this one.

Or what about saying at the end: A basic kind of problem we will want to understand is whether a given vector is a linear combination of some other vectors. As we will see, this question is closely related to the problem of solving linear systems!

Feedback on solution-sets

In reference to version cebba0d48f43784542dd70c9421777691a9419c6

The last subsection starts with: ...we have now associated two completely different geometric objects, both described using spans.

But we haven't defined column span yet, so I am confused.

Feedback on row-reduction

In reference to version f6ae88a

4 could be rephrased as: Each pivot is equal to 1. Seems simpler a little...

Feedback on solution-sets

In reference to version cebba0d48f43784542dd70c9421777691a9419c6

In the key observation, let's not use the term "nonempty".

Feedback on systems-of-eqns

In reference to version 79e5138

We talked about moving the last part of the section to 2.2. What about leaving the definition of inconsistent in 2.1? I think that's helpful. It divorces the concept of inconsistent from the matrix stuff.

Feedback on solution-sets

In reference to version cebba0d48f43784542dd70c9421777691a9419c6

In the first recipe it is unsettling that the vectors seem to go on forever.

Feedback on solution-sets

In reference to version cebba0d48f43784542dd70c9421777691a9419c6

After the recipe we need an un-knowled example of parametric form.

Feedback on solution-sets

In reference to version cebba0d48f43784542dd70c9421777691a9419c6

In the beginning of the inhomogeneous section, it says that p is a solution to Ax=0, when it should be Ax=b.

In general, when an example is referred to immediately afterward, it probably should not be in a knowl. This means that it is part of the narrative.

REF conventions

Tell the reader in 2.1 that "a matrix has a pivot here" means it does so after REF.

Then remind about the convention the first couple times it comes up.

Feedback on vectors

In reference to version a327fdb

I think we've discussed this before, but the subscript p feels unnatural to me. Shouldn't it be k?

Feedback on spans

In reference to version a327fdb

We say there are two ways of thinking about linear systems: (1) augmented matrices and (2) vector equations. Why not say we have three ways of thinking of the same idea: (1) linear systems, (2) augmented matrices, and (3) vector equations. We should also emphasize that the reader should get comfortable going from any of of these forms to any other.

Again, happy to make edits. I'm a little uncomfortable making such changes without an ok.

Feedback on parametric-form

In reference to version a327fdb

I don't know if I feel strongly about this, but R2 = R2 + R1 seems weird to me. I guess it's sort of a computer programming notation? Should we point this out at least? In class I use R2 --> R2 + R1.

Feedback on row-reduction

In reference to version 79e5138

I feel like we should hint towards free variables at the end. Maybe say that if all the columns are pivots, then you basically have the answer. But what do you do if you have some columns without pivots? What is the solution?

Feedback on vectors

In reference to version a327fdb

There is a blue box that confuses me:

Unless otherwise specified, we will assume that all vectors start at the origin.

WE just got done saying that the great thing about vectors is that they can be located anywhere. And now we are saying they're basically always going to be at the origin. What is really meant by this blue box?

Also, it would be good to give an example of why vectors are useful. Maybe the velocity of a moving object?

Feedback on vectors

In reference to version a327fdb

The word "element" is now used 4 times. I say we just kill it completely. I think it's already implicit that elements of R^n are points. And now we are saying that we can also draw them as arrows.

By the way, happy to help make changes. But since this one could be controversial, I'm using this forum to check with you.

Feedback on matrix-equations

In reference to version cebba0d48f43784542dd70c9421777691a9419c6

In "Matrix Equations and Vector Equations" we say X is equivalent to Y and conversely Y is equivalent to X. Isn't this redundant?

Feedback on systems-of-eqns

In reference to version 79e5138

suggestion. replace

by a single piece of data in some R^n

with

by a single piece of data: a point in some R^n

Feedback on row-reduction

In reference to version f6ae88a

The first example knowl should have a couple of more typical examples, one not reduced, one in REF, and one in RREF.