Difficulty in teaching the coordinates of a vector with respect to a basis $\{v_1,v_2,\ldots,v_n\}$

Question

Let $V$ be a finite dimensional vector space and let $B=\{v_1,v_2,\cdots,v_n\}$ be a basis of $V$. If a vector $v$ can be written as

$$v=a_1v_1+a_2v_2+\cdots+a_nv_n,$$

we call $(a_1,a_2,\cdots,a_n)$ the coordinates of $v$ with respect to the basis $B$.

However, I find teaching this definition contradicts our teaching that in a set, the order of elements does not matter. Here if we write $B$ as $\{v_2,\ldots,v_n,v_1\}$, it is still the same set; however, the coordinates of $v$ will usually be different.

So in teaching the coordinates with respect to a basis, should we introduce the notion "ordered basis" (as an ordered $n$-tuple of vectors) and define "the coordinates with respect to an ordered basis" instead? I find that in most Linear Algebra textbooks that I am aware of, this issue is never raised.

Answer 1

At the level of standard linear algebra courses, where we consider bases only for finite-dimensional vector spaces, we should work with ordered bases, so that the coefficients of a vector come with an ordering (and thus constitute a vector in $\mathbb R^n$ or $\mathbb C^n$) and so that the matrices that represent linear transformations come with their rows ordered and their columns ordered (and might therefore be written on a page).

At a more abstract level, we could use unordered bases. The coefficients of a vector $v$ would then not be a sequence of scalars but rather a function from the basis into the field of scalars --- the function assigning to each basis element $b$ the coefficient of $b$ in the expansion of $v$. Similarly, transformation matrices would have their rows and columns not ordered but rather indexed by basis vectors. The point of this more abstract approach is that an ordering would be used only to match up which coefficient goes with which basis vector (or, in the case of matrices, which pair of basis vectors), and so we can forget the ordering and work directly with the matching.

I have not tried to teach the more abstract approach in linear algebra classes, and I'm pretty sure I don't want to try. In classes I'll stick with ordered bases.