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.

  • $\begingroup$ Just to point out that your example is false. $S\cup \{x\}$ is still linearly dependent because the linear dependencies from $S$ still hold. $\endgroup$ – Kevin Carlson Sep 26 '14 at 3:31
  • $\begingroup$ Sorry, I meant that $S$ is linearly independent to begin with. Corrected now. $\endgroup$ – Train Heartnet Sep 26 '14 at 3:35
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    $\begingroup$ They all use lists. How can you even define linear (in)dependence without numbering the vectors at stake? –Another point: If $a$ is a nonzero vector then the set $\{a\}$ is linearly independent, but the two vectors $x:=a$ and $y=a$ are linearly dependent, even though $\{x,y\}=\{a\}$. $\endgroup$ – Christian Blatter Sep 26 '14 at 8:57
  • $\begingroup$ Unless I'm missing something, linear (in)dependence can be defined without ordering the vectors: A set $\{v_1,v_2,...v_m\}$ is linearly independent in $V$, if the only representation of $0$ in terms of a linear combination of $\{v_1,...v_m\}$ is $0=a_1 v_1+a_2 v_2+....a_m v_m$ with all $a_i = 0$. A set is defined to be linearly dependent if it is not linearly independent. There is a numbering of vectors, but no ordering of them. But yes, your second point is valid and Axler does raise this. $\endgroup$ – Train Heartnet Sep 27 '14 at 7:08

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.)

  • $\begingroup$ Ah, I see. Thank you for clearing that. :) $\endgroup$ – Train Heartnet Sep 27 '14 at 8:43

I've also read and enjoyed Axler's book, and also wondered about his explicit use of lists rather than sets of vectors. Every other book I have look at based the main arguments on sets of vectors, rather than lists. Two properties that distinguish a finite list from a finite set are: (1) A list is an ordered collection, in which each object is assigned a position, and (2) A list can contain more than one instance of an object, whereas a set is usually defined as an unordered collection of distinct objects. Specifically, the definition of a list of vectors does not seem to exclude the possibility that two or more vectors in the list are equal. If a list of vectors is known to be linearly independent than, of course, it cannot contain multiple copies of the same vector, but may arguments in linear algebra involve statements about finite lists of vectors that are not necessarily linearly independent. The type of collection that is obtained by ignoring the ordering of a list is sometimes referred to as a "multi-set", that is, a collection in which each object in the collection is assigned a multiplicity. Though other texts generally refer to a set of vectors in places where Axler uses a list (e.g., to refer to a finite list that is a linearly independent list or a spanning list of some space), I have wondered whether it might not be more appropriate (or at least more general) to refer to a multi-set (i.e., an unordered list). It appears to me that, for clarity, a development of the subject that is based entirely on sets, rather than lists or multi-sets, should be more explicit about the implications of the requirement that a set not contain multiple copies of the same vector than is usually the case. It thus appears to me that Axler's use of lists rather than sets is a pedagogical advantaged because:

1) The implicit requirement that a collection of vectors not contain multiple copies of the same vector in any discussion based on sets is an unnecessary complication that is avoided by the use of either lists or multi-sets.

2) The introduction of an explicit ordering is often convenient, particularly when discussing components of a vector defined relative to a specific basis or indexing of coefficients in a linear combination, which require introduction of some sort of indexing if only to simplify notation.

The main disadvantage of the use of lists rather than sets of vectors seems to be that the student has to keep in mind that some important properties of a list, such as linear independence or the span of the list, are actually independent of order in which vectors are listed. This, however, seems to me to be quite easy for student to understand and keep in mind.

I've been thinking about this because I am writing my own notes for a math class that is aimed at engineering and physical science students, rather than undergraduate mathematics majors or graduate math students. I thus far think I prefer Axler's approach based on lists for reasons of clarity, but would be interested in seeing other comments on this.

  • $\begingroup$ I use the same approach with lists, which can be seen as a generalization of matrices: a matrix corresponds to the list of its columns, and from the list, rather than the set, we can reconstruct the matrix. $\endgroup$ – egreg Jul 11 '18 at 14:46

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