Here's a rather mundane example: a basis of a vector space. A basis is usually defined to be "a set of vectors such that...." The problem with this is the following: $$\begin{bmatrix}0 & 0 & 1 \\ 1 & 1 & 0\end{bmatrix}$$Do the columns form a basis for $\mathbb{R}^2$? The answer is "yes" if a basis is a set of vectors...but this is obviously false.
The same applies to the usual definition of linear independence. Are the columns of the following matrix linearly independent? $$\begin{bmatrix}1 & 1 \\ 1 & 1\end{bmatrix}$$The answer is apparently "yes," since in my experience linear [in]dependence is usually defined only for a set of vectors, and the set of the columns of that matrix consists of a single nonzero vector...again, obviously false: the columns of a square matrix shouldn't be linearly independent unless the matrix is invertible.
In my mind, the collections of vectors to which pretty much all linear algebra concepts apply are tuples of vectors, not sets. This stems from the fact that the fundamental operation on a vector space is the linear combination, which operates on a tuple of vectors and a tuple of scalars (indexed by the same set). If I want to figure out whether a collection of vectors spans a space or is linearly independent, the next thing I'm going to do is consider a linear combination of those vectors. Therefore, it only makes sense for that collection to be something to which "linear combination" readily applies, namely a tuple! Still, I've seen reputable textbooks define everything only in terms of sets.
