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.