I've just been reading through the first few pages of 'Linear Algebra Done Right' by S. Axler and so far, the concept of a 'list' seems a lot like a vector. Are they one and the same? If so, is there a reason why Axler chose to call it a list? Perhaps maybe to rid the idea that a vector is a 'directed line segment'? Thanks
I don't have that book so cannot see exactly how he defines "list" but, at least according to any definition I would consider reasonable, no, they are not the same. I would take a "list" to be an ordered sequence. I can, have, for example, my grocery list: bread, milk, tuna fish, green beans, potatoes. That is pretty much equivalent to the "list" data type you will find in Pascal, C++ or Java. Now, while there are many different, more or less equivalent, ways to define "vector", they all give an algebraic structure. We must be able to add two vectors and multiply a vector by a number (or, more generally, an element of a field). That is certainly NOT the case for the "list" above. If he is saying "a list of numbers with defined addition and scalar multiplication", that would be the same but I can see no good reason for the non-standard terminology.
No. He says so himself: In general, a vector space is an abstract entity whose elements might be lists, functions, or weird objects. However you learn abstract linear algebra, that a vector is a 'directed line segment' is one of the first notions you have to get rid of.
No. A list's components need not necessarily be objects that obey the rules that define a vector (see page 9). For example, a list of positive integers has no additive inverse, ie there is no positive number w such that (v) + (w) = (0), whereas a vector does have an additive inverse. Perhaps, although he does mention the traditional concept of an arrow as a vector on page 6 ("when we think of x as an arrow, we refer to it as a vector") so I think not. I think it's more likely to be because it conveys the idea of an n-tuple (see the side note on page 4) better than "n-tuple" does. (Thank goodness for "Look Inside" on Amazon! )
As I recall, he uses lists to group objects much as you would do with a set, but with two differences: first, the order in which the elements appear is significant, so (1,2) is not the same as (2,1); and second, there can be repetitions, so you can have a list like (2,2) which is distinct from (2), something you can't do with sets.