Are 'lists' and vectors the same thing?

autodidude

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

HallsofIvy

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.

D H

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.

NemoReally

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! )

jbunniii

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.

autodidude

Makes sense now, thanks a lot of everyone!