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

Entry: What is Linear Algebra? - A87849913
Author: SashaQ - eccentric editor - U9936370

I've been rummaging in the Flea Market again

Original Entry: A893900 Linear Algebra
Original Author: U166086 GTBacchus

A fine attempt, but I suspect you haven't quite made it simple enough for the non-mathematician to appreciate.

e.g. (on describing matrices) "If that doesn't make sense, don't worry. The important thing is that a matrix represents a transformation, and that doing one transformation, and then another, is like multiplying the matrices together - in the right order!"

I'm pretty sure that if they didn't understand the matrix, they won't understand the transformation analogy either.

What it needs are real-world practical examples of this theory, and ideally a few diagrams. I have some good books on the subject, and I'll be happy to look up a few ideas, if you'd like.


And finally,

"The determinant of B, the flipping matrix, is -1,"

I know it's a frustrating discipline at times, but isn't there a more descriptive word?

Thanks for reading Icy

Yes, diagrams would be good, as eg it wasn't easy to understand the motions of the book in the original Entry, and I'm not sure I've made it any clearer...

I've tweaked the Entry a bit. I did think that the original Entry had quite good logical progression, defining a term and then using the term, so people could follow the transformation analogy, but people's opinions of that are welcome.

What kind of thing did you have in mind in relation to real-world practical examples?

Let me do some reading first. My real-world examples are generally things like dividing cakes or chocolate...

OK, I have to take back a lot of what I said. I can't see a particularly easy non-mathematical analogy for systems of linear equations nor matrices nor eigenvectors. Unless you want to do a picture of three apples and two oranges costing 85p, alongside four apples and three oranges costing £1.20 (say), then using it as a worked example.

Transformations, however, are pretty easy to do. I have a diagram on these for my wallpaper patterns entries (I'm in the process of redrawing it as it's the wrong shape).

You just need a diagram showing a shape, then its reflection, rotation and shear, for example, with associated matrices.

Thanks Icy

I understand your point, where your Entries have "here is a problem, and this is the maths" but this one is "here is some maths, which is useful for solving some problems"

I shall tweak the Entry a little bit more and sort out some diagrams

Hi SashaQ

I hope you don't mind mind, I am really trying to understand this Entry.

What makes these linear is that they don't have any funny stuff like x2, or the cosine of x, or any logarithms or square roots, or anything but multiplication by ordinary numbers, and addition and subtraction.

In your first example 5x=45 you use division, shouldn't this be added to the above statement?

Well, addition is commutative, because x + y is always the same thing as y + x. The order of the terms doesn't matter. If A and B are matrices, on the other hand, then AB is not usually the same as BA. Multiplying them in one order is different from multiplying them in a different order.

Actually A x B always equals B x A, If you add a third unknown you are absolutely correct, you can do column addition in any order, you can not do column multiplication, subtraction or division.

As I understand it linear algebra produces either a single point, or more usually a straight line as its answer. This might be better stated in the opening section, before giving the fuel tank illustration.

I also wonder if the answer must be even, like in elementary school, how about 5x=42 - x=8.4?

This might be clearer if you added a few more example and introduce a third variable.

The matrix is a little confusing, could you give us a few examples about how the numbers in a matrix can be manipulated?

I use a good deal of maths in my work, but just for practical purposes, like figuring out how far a post can span without bending, or telling the people in the how long each piece needs to be cut before assembly.

I hope this helps, I think this will be a great addition for the Guide

F S

Probably the most confusing thing about matrix multiplication is that they call it 'multiplication'. It's not the same as arithmetical multiplication at all. It's a multidimensional operation involving the sum of the products of different combinations of the cells, and if you multiply the matrices in the reverse order, the result is generally different.

I'd love to give you a simple non-mathematical analogy, but I'm struggling to find one.

Linear transformation is probably the best way to visualise this.


If we rotate a shape by x radians anticlockwise, then the matrix to describe this is

| cos(x)-sin(x) |
| sin(x)cos(x) |

What this means is that if you have a point on the shape at coordinate (a,b) before rotation, then after rotation it will be at point

| cos(x)-sin(x) || a |
| sin(x)cos(x) || b |

i.e. you multiply the matrices.

If you then rotate it further by y radians, then it will be at the point

| cos(y)-sin(y) || cos(x)-sin(x) || a |
| sin(y)cos(y) || sin(x)cos(x) || b |

Multiplying the matrices again.


Simples

Thanks both

I've made some changes, so I hope I've answered most of your queries, FS. Thanks for reading - much appreciated

I did battle with GuideML today too, but not very successfully, so I'll look at that again tomorrow...

I've always been stupid at maths , so I read this with trepidation.

I think I more or less understood. So

Thanks for straightening that out.

One confusing thing I see remaining is you still tell us that the cosine is not used, but Icy's second example uses the cosine to turn the object. Perhaps this one line should be removed, even if it means losing the link.

Does 'multiplication' simply mean manipulating the numbers? (maybe a footnote?)

F S

SashaQ's example was for a specific rotation of p1/2 radians, so you can plug in the numbers into my formula to get ones and zeros.

But remember too that Sasha's example rotates clockwise (mine was anticlockwise), so you'll have to reverse signs.

pi/2 that should read, not p1/2

Thanks everyone

I added a footnote on 'multiply' and I'm still doing battle with the GuideML.

I've left the special case of 90 degree rotation in the Entry, so as not to go into too much detail in relation to examples...

~

Icy's example has unfortunately used x to mean some number, and the x from the linear equations is called a.

So the equations in this case are cos(some number)x - sin(some number)y = 0 etc, so they have the same format as 2x - y = 7, but eg the 2 is replaced by the number that is cos(some number).

I hope that helps...

Thanks Sasha.

And I wasn't trying to fit in with your conventional notation - ignore mine and write it any way you like.

Gah - I ordered a book from the library that looked as though it would be helpful, and I found out today that it is lost, so it may take some time for me to get it, if I get it at all...

The book turned up

Small tweaks have been made to the Entry, and diagrams will appear soon

Updated