Linear independence and grammar

Question

Let $A$ be a commutative ring.

Normally, the linear independence is a property of a subset or a family elements of an $A$-module. But one often sees statements like

"$v$ and $w$ are linearly independent."

Although there is no possibility of confusion, I was wondering if this is strictly correct?

Edit: Regarding $v_1,v_2,v_3$ as a family, statements like

"$v_1, v_2, v_3$ is linearly independent."

seem rather odd.

Answer 1

I found the following "definition" in Hoffman's Linear Algebra (2ed.):

"If the set $S$ contains only finitely many vectors $\alpha_1,\ldots,\alpha_n$, we sometimes say that $\alpha_1,\ldots,\alpha_n$ are linearly independent instead of saying that $S$ is linearly independent."

To be extremely pedantic, this definition (or convention) still has an issue (If $v$ is a non-zero vector, {v} is linearly independent but $v$ and $v$ aren't). So the following modified version for a family of elements seems the safest:

For a finite family $F=(m_i)_{i=1}^n$, we sometimes say that $m_1,\ldots,m_n$ are linearly independent instead of saying that $F$ is linearly independent.

Answer 2

It is an abuse of language. It means that the set $\{v, w\}$ is linearly independent.

Since I took my time to answer here, might as well add something: there is a difference between sets and sequences. Notice that $\{v, v, w\} = \{v, w\} = \{w, v\}$, but $(v,w)$ has just one meaning. The set $\{v, v, w\}$ is linearly independent. The sequence $(v,v,w)$ is linearly dependent, while $(v,w)$ is linearly independent. But this is just a technical issue, most likely confusion will never arise.