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?
Best Answer
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