Linear Algebra and Geometry by Kostrikin and Manin: Remark regarding diagrams and graphic representations.

Question

On page 5 of this book there is a particular section of the book that I am having trouble trying to understand as to what the authors' are trying to point across. It is concerning linear algebra. I will place in bold the parts I need additional explaining and number them as (1),(2),(3) which will be associated with the numbered questions below.

So it begins like this:

Remarks regarding diagrams and graphic representations. Many general concepts and theorems of linear algebra are conveniently illustrated by diagrams and pictures. We want to warn the reader immediately about the dangers (1) of such illustrations.

a)Low dimensionality. We live in a three-dimensional space and our diagrams usually portray two- or three-dimensional images. In linear algebra we work with space of any finite number of dimensions and in functional analysis we work with infinite-dimensional spaces. Our "low-dimensional" intuition can be greatly developed, but it must be developed systematically(2). Here is a simple example how are we to imagine the general arrangement of two planes in four-dimensional space ? Imagine two planes in $\mathbb{R}^3$ intersecting along a straight line which splay out everywhere along this straight line except at the origin, vanishing into the fourth dimension(3).

1) How does the particular example above show the dangers of such an illustration?

2)The author didn't elaborate much on this point. What does it mean to develop such intuition systematically and how?

3) I'm not quite sure what the author is trying to say about this, and with no pictures in the book it is quite difficult for me to figure out what it trying to be put across by the authors. If someone could try to explain so that I can have a mental "picture" in my head what is actually intended by the author. Diagrams and pictures accompanying an explanation would also be greatly appreciated (though one is not obligated to provide one.)

NB: I guess part of the reason why I don't fully capture what the author is trying to get across is because I can't quite get my head around the example about the two planes in four dimensional space.

Answer 1

I'm not sure that what I'm going to say is exactly what the author meant. However, I'll try to explain a little of what I understand about this. First, the dangers is that sometimes when you try to take your three-dimensional intuition to higher dimensions it can break down. For instance: imagine two planes that are not paralel in three dimensional space, and imagine that I ask you if the planes will intersect. You'll tell me that they will.

That's fine. Now I ask the same question about two planes at the $4$-dimensional space. If you tell me that they'll intersect you're going to make a great mistake. There's a cool analogy: two lines that are not paralel in $2$-dimensional space will always intersect, but in three dimensions, you can have two lines that are not parallel, however each of them at disjoint planes. In this case there won't be intersection. The same is with planes in $4$-dimensional space, each of them can be in disjoint $3$-dimensional subspaces.

About intuition is something like this: after you work a lot with those things - vectors in $n$-space, hyperplanes, subspaces generated by specific sets - you'll start to get some intuition, you'll start to have your own way to understand those things, because you get used to them.

The diagrams and figures are of course bound to two or three dimensions, however as I've said, when you work a lot with those things, you'll be able to take the intuition behind the diagrams and pictures to $n$-space. It'll become so natural, that it'll be like if you could "see" the $n$-dimensional space, because the notions defined there are just developed generalizing and sharping the definitions in two or three dimensions (which you can see and intuit by the diagrams and pictures).

In summary it's all about understanding that notions in $n$-space generalize notions in $2$-space and $3$-space and getting used to it. When you can map your low dimension intuition to any dimension, you'll see that everything makes sense, you'll look at those definitions and think that they're as natural as the definition of a line in $2$ dimensions.

I hope this clarifies a little what you're troubled with. And also, this is just my point, I cannot tell you that this is exactly what the author meant, but I hope it helps you.