[Math] Lists versus sets in linear algebra

linear algebrasoft-question

I’m currently learning linear algebra from “Linear Algebra Done Right” by Sheldon Axler. The author, in his proofs, makes use of lists of vectors, as opposed to the more conventional usage of sets of vectors. I have some questions concerning this:-

  1. Are lists standard in linear algebra? I mean, I have referred to many other books, and all of them seem to use sets rather than lists.

  2. If I study linear algebra at a higher level, will my current study using lists prove to be a hindrance? In essence, I mean to ask whether results in higher linear algebra texts make use of lists or rather sets, in their proofs?

  3. Aren’t certain results more cumbersome to prove using lists? As an example, consider this statement: If $S$ is a linearly independent set in $V$, and $x \notin span(S)$, then prove that $S \cup \{x\}$ is linearly independent. This is fairly easy to prove considering $S$ as a set. But when it comes to an analogous result for lists, since the order matters, wouldn’t there be many possibilities of adjoining $x$ to the list? And for each of these possibilities, wouldn’t I have to prove that the list is linearly independent? If $x$ is adjoined at the very end of the list, this follows easily from the linear dependence lemma, but what if $x$ is adjoined to the list at some arbitrary position? Wouldn’t statements like this, which involve adjoining (or equally, removal) of vectors be more cumbersome to prove, in the case of lists?

  4. Are there other disadvantages of using lists over sets? Frankly, I am in love with Axler’s book and his simple, clean proofs using lists, over other linear algebra texts at this level. But I’m worried that this very simplicity is going to prove troublesome when I decide to study linear algebra at the graduate level.

I apologize if this isn’t the right place to post this, but I haven’t seen any discussion regarding this anywhere else, and I thought someone here might be able to give insights.

Best Answer

There's not a problematic difference. What a list is, in terminology you're more likely to encounter in the future, is simply an indexed set, which in general is just a set $S$ endowed with a function $i:I\to S$, but which is usually indexed by $\{1,...,n\}$ or by $\mathbb{N}$ in the familiar way. So, the difference is that a list comes with a specified order, as a set does not. Having this order built in is important for the theory of orientations and determinants, whereas it's never important not to use a list. Indeed, every set is canonically indexed by itself (via the identity function,) so the study of lists is in this sense strictly more general than the study of sets.

For your example, you would probably just prove the proposition in case $x$ is adjoined to the end of the list. There's an easy theorem, which Axler may or may not point out explicitly, that the span of a list doesn't depend on the ordering, i.e. it's just the span of the underlying set, so this covers concerns with adjoining $x$ at different points. (The theorem is proven via the commutativity of vector addition.)

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