h2g2 The Hitchhiker's Guide to the Galaxy: Earth Edition

Henry, thanks. I've fixed those technical points, I think:

(Systems containing zero equations are not very interesting.)

It was a throwaway line anyway. I threw it away. I'm not even sure that a "system of equations" is defined so that it can have zero equations in it.... *checks book* No. According to one book anyway, it's "a collection of one or more linear equations..."

---------

Generally though, if you have three unknowns, you'll need three equations, and so on.

Henry: "I'm never sure if 'generally' means 'in general, and in all cases' or 'usually'. Usually you do need 3, but you might be unlucky and get linearly dependent ones. Not sure how to phrase this so its precise but not unweildy."

I redid that bit, and part of the next paragraph, too. 'Generally' is a problem word. Non-mathematicians use it to mean that something isn't *always* true, and mathematicians use it to mean that something is.

---------------

Henry: "Book flipping example is good. Could maybe explain a little more what the flip is. It might not be obvious that youre supposed to do the same hand motion whatever orientation the book is in. Maybe talk about it rolling away from you, or spit roasting it with the skewer parallel to your chest."

I think I've made it clearer.

-------------------

a square matrix has as many eigenvalues as it has rows (or columns).

Henry: "unless they are repeated..."

Good point. Noted. Clearly, I hope.

--------------------

When a matrix has complex eigenvalues, they occur in complex conjugate pairs,

Henry: "I'm sure thats true if the matrix has real entries, but don't think its true if it has complex entries."

Very good point! I haven't studied matrices w/ complex entries, but what you say is obviously true. That sentence is gone now. It wasn't that essential anyway.


GTB

Pencil Queen, hi.

"I wonder about putting the vector part first but only because I remember vector maths so comparing vectors to matrices would make more sense than explaining matrices and then comparing them to vectors(?)"

I know what you mean... but I'm sort of following the format of the class I took. Systems of equations are a familiar thing to start with, and they lead naturally into matrices. I'll be doing an entry on how to solve linear systems w/ matrices, and I'll link to it. Later, vectors come up in connection with vector spaces, which are bizzare, and not usually introduced until you have some familiarity with matrices.

(I've actually got about half a dozen entries planned now, which I'll either put through PR all at once, or put into PR for a week, and then send them to Uni, or something. How to get Uni entries reviewed is still an unresolved issue...)



"I think if you want to make it a bit more accessable you should mention what the curves look like from looking at the equations (m=slope (-ve down, +ve up), c=intercept (where the line crosses the vertical axis)"

That's how I learned to graph lines too. You graph the inconsistent equations and see that they're parallel. Only problem is, what if your inconsistent equations are:

x + y + z + w = 4
2x + 2y + 2x + 2w = 5

Those are parallel hyperplanes in 4-space, but nobody's going to see that by graphing it. You've got to use matrices.

Your point is totally valid though. Slope-intercept equations are familiar. I'll find a way to mention them, and to say that you have to move past graphical representations when you have more than 2 or 3 dimensions.

An entry on graphing lines in the xy-plane would be a great one to have, but it's not part of this project.



"I think for each example you need an example equation from a RL situation, just talking about equations without example applications seems a bit pointless, especially when the techniques are used in practical subjects like engineering."

I'll probably use that suggestion more in the entries on more specific topics. Once they're written, I'll be revisiting this one. I should go work on that now. I'll definitely find out how to get diagrams into those.


Thanks for the comments!

GTB

You say you normally only do linear algebra in the 3rd year, but at Hull Uni we do it in the first year (well, applied linear algebra, i don't know if it's the same thing or not cos i'm doing it next year (second year) due to doing maths/french not just maths)...

Well, now that you mention it, I'm in no way qualified to make sweeping statements about what goes on at most universities and colleges in this world. What was I thinking? Perhaps I should do a survey and find out when people in the UK and US who study Linear Algebra tend to run into it. If I get around to it, I'll ask at the h2g2 Maths Lab. Got a few other things to do first though...


GTB

In the UK Linear Algebra is usually a first year course on a straight Maths degree - and usually part of the Maths that gets taught to Physicists and Engineers in their first / second years.

In fact, it is also true that various parts of Linear Algebra are on the A-level Further Maths syllabus for at least some exam boards - I certainly learnt how to multiply and invert matrices before going to university.

Regarding this entry (A893900), I think that it can be split into two. First, an entry on matrices (maybe you could add more to Matrix Multiplication - A895511 ?) describing how to add, multiply, find the determinant, invert, and so on... and then this entry on Linear Algebra wouldn't seem so long since you would only need to refer to the first entry.

Does that make sense?

I like that suggestion, Andy, about dividing up the entry. I don't know when I'll actually do it..... and I do have one question.

I've seen matrix multiplication presented two different ways. One defines it where each element is the dot product of a row from the first matrix with a column from the second. I think this is the most common way of presenting it, but I don't know how intuitive it really is. It seems contrived until you get used to it.

The other approach that I've seen is to define the product, Ax, of a matrix with a column vector first, that the product is a vector that's a linear combination of the columns of A, using the entries in x as weights. This definition is then extended to matrix multiplication: each column of AB is the product of A with the corresponding column of B.

I think that the second way makes more sense, and helps with ideas like column space, but I don't know whether it's very common, or if people will think it's unfamiliar.


Does that make sense?


GTB

I learnt it as "row into column". The element at row r, column c of the result matrix is row r of the first matrix multiplied by column c of the second matrix. To multiply them together, multiply all the individual values and add them up.

Did this ever get anywhere, or do you still want comments on it?

It needs, at the very least, a major overhaul. I don't quite see myself working on that very soon.

I'll still take comments, sure, and I'll continue to beleive that I may someday revisit this entry.



GTB

You could always take it out of the writing workshop until you have time to work on it?

Done, and done.