Should I use set notation or list notation when writing out a basis of vectors? I think in Sheldon Axler's Linear Algebra Done Right, he makes a comment about why the technically correct way is to write vectors in lists, such as $(v_1, ... v_n)$, while many books use set notation, such as $\{v_1, ... , v_n\}$.
I believe set notation just includes the distinct vectors, while lists allow repeat vectors, such as this list $(v_1, v_2, v_2, ..., v_1, v_n)$.
Or is it not important and both are acceptable?
 A: Opinions on this issue differ, but I strongly believe that a basis (particularly in finite-dimensional linear algebra) should be a list, not a set. Here I am using "list" to mean the same thing as "ordered set". Here are two reasons why using sets does not work well:


*

*It is often convenient to talk about the matrix of a linear map $T \colon V \to W$ with respect to a basis of $V$ and a basis of $W$. However, if the basis is a set, then it makes no sense to talk about, for example, the first column of this matrix. If the bases are lists, then the first column makes sense and is well defined.

*If $v_1, v_2, v_3$ are vectors in a 2-dimensional vector space $V$, then the list $v_1, v_2, v_3$ is for sure not linearly independent (no list of length 3 is linearly independent in a 2-dimensional vector space). However, if one works with sets, then it is not for sure that $\{v_1, v_2, v_3\}$ is not linearly independent because it may happen that $v_3 = v_2$, in which case  $\{v_1, v_2, v_3\} = \{v_1, v_2\}$.
--Sheldon Axler
A: I agree with Professor Axler that a basis should not just be a subset; in particular, it should be indexed by another set, call it $I$. However, I disagree with the general contention that $I$ should be required to carry a distinguished order.
Let me elaborate.
Firstly, here's my preferred definition of the term "basis":

Definition. Let $V$ denote a vector space.
Then a family in $V$ consists of a set $I$ together with a function $e : I \rightarrow V$. A basis of $V$ is a family $(I,e)$ in $V$ such that all $x \in V$, there exists a unique finitely-supported function $a : I \rightarrow \mathbb{R}$ satisfying $$x = \sum_{i:I} a_i e_i.$$

A few comments:

*

*This allows repeated elements, in principle at least. You can then get students to prove that if $(I,e)$ is a basis of $V$, then $e$ is always injective (i.e. there are no repeated elements.)


*If $(I,e)$ is a family in $V$, then there is a unique linear transform $$\mathbb{R}\langle I \rangle \rightarrow V$$ extending $e : I \rightarrow V$.
It can be shown that the aforementioned linear transform is

*

*injective iff $(I,e)$ is a linearly independent family

*surjective iff $(I,e)$ is a spanning family

*bijective iff $(I,e)$ is a basis.



*There's no need to assume that $I$ carries an order for any of this to work.


*An $m \times n$ matrix can be defined as a linear transform $\mathbb{R}^m \leftarrow \mathbb{R}^n$. But in fact, this can easily by generalized by saying that an $I\times J$ matrix is a linear transform $\mathbb{R}\langle I\rangle \leftarrow \mathbb{R}\langle J\rangle$. This basically amounts to labelling the rows of the matrix with the elements of $I$ and the columns of the matrix with the elements of $J$, rather than relying on the order to induce a canonical "naming system" on the rows and columns.
Of course, the more computational you get, the more that orderings begin to matter. Gaussian elimination, for example, requires that the indexing set carry a distinguished order. So define:

Definition 1. Let $V$ denote a vector space.
Then an ordered family in $V$ consists of a family $(I,e)$ in $V$ together with a well-order on $I$. An ordered basis in $V$ consists of a basis $(I,e)$ of $V$ together with a well-order on $I$.

I think these conventions are basically optimal.
