Cartesian Products

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jpanknin

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I'm going through Serge Lang's Intro to Linear Algebra. In one section he describes the Cartesian Products of higher dimensions than just R x R (see image below). I can't find anything that shows what this looks like geometrically, at least for R^3. The last sentence in the image says "This means that we view separately the first two coordinates (x1, x2) and the last two coordinates (x3, x4)." This seems to me that (x1, x2) and (x3, x4) are just two points in R2 rather than a single point in R4. I'm having trouble visualizing this geometrically. Also having trouble understanding how something like R2 x R = R3 ==> (a, b) x c would result in a single point in R3 instead of something distributive like (ac, bc). Any help or pointing to resources online would be appreciated.

1728060217947.png
 
A point in R^4 is represented by an ordered 4-tuple [imath](x_1,x_2,x_3,x_4)[/imath]. The point is that we could view that as a point [imath]((x_1,x_2),(x_3,x_4))[/imath] in R^2 x R^2, that is, a pair of pairs.

If we want to view R^4 as R^3 x R, then we are seeing such a point as [imath]((x_1,x_2,x_3),x_4)[/imath], a pair consisting of a triplet from R^3 and a single real number from R.

Distribution applies to a product of sums, not to a product of products.
 
Dr.Peterson said:
A point in R^4 is represented by an ordered 4-tuple [imath](x_1,x_2,x_3,x_4)[/imath]. The point is that we could view that as a point [imath]((x_1,x_2),(x_3,x_4))[/imath] in R^2 x R^2, that is, a pair of pairs.

If we want to view R^4 as R^3 x R, then we are seeing such a point as [imath]((x_1,x_2,x_3),x_4)[/imath], a pair consisting of a triplet from R^3 and a single real number from R.

Distribution applies to a product of sums, not to a product of products.
Click to expand...
Ok, but how would [imath]((x_1,x_2,x_3),x_4)[/imath] become a single point in R4? Would it just essentially be [imath](x_1,x_2,x_3,x_4)[/imath]? How a point goes from [imath]((x_1,x_2),(x_3,x_4))[/imath] to [imath](x_1,x_2,x_3,x_4)[/imath] is where I'm not understanding.
 
It's not that those really are the same thing, but that they are equivalent in some sense; every point in one form corresponds to one point in the other. I'd have to see more context to be sure how precisely he is explaining things, and what he is going to say "later". Has the book mentioned isomorphism yet? My impression is that this is just an informal introduction to these ideas, and an informal way to think about them.

I see the idea as an extension of the fact that you can see R^3 as an xy plane with a z dimension added, so each point in space can be thought of as a point on the plane, together with an "altitude: (x,y,z) "=" ((x,y), z).
 
It has not mentioned isomorphisms yet. That's pretty far down the road in this book. He doesn't really explain it at all in the section from the screenshot, just mentions it briefly. So I think it is just an intro to what's coming later.

R^3 makes more sense to me than a R^2 x R^2, but hopefully it'll be addressed "later." Appreciate the help.
 
I found the book in archive.org, and I see that this is on page 4, where he is just introducing the idea of more than 3 dimensions, with reference to an article by d'Alembert where he happens to use the idea of a product of spaces. Nothing has yet been formally defined.

1728083438852.png​

So, yes, you can expect that anything he works with formally will be explained better than this.
 
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