My Mathematics Pet Peeves

Mathematicians aren't the greatest writers in the world. Their job to convey very dense, structured information in the most efficient and concise way possible. Moreover, their audiences can be highly technical and knowledgeable. Too many details bog down the discussion while adding no clarity. And what may be obvious to one person may be inscrutable to another.

I understand all that. But there are some avoidable pet peeves that really bother me. Here are some of them. Let me know what you think!

"It is Clear"

This is used in two contexts. In one case, it's used when something really is clear (e.g., "Since n is an integer, it is clear 2n is even"). Adding "it is clear" adds nothing, so just say "2n is even." Or better yet, don't even bother saying it because it's truly obvious in your argument. In the other case, the conclusion isn't clear at all except to the most superhuman of minds. Some additional discussion supporting your conclusion would be helpful.

"Linearly independent vectors"

This one bugs me. There's no such thing as a linearly independent vector, or even several linearly independent vectors. There can, however, be a set of linearly independent vectors. Linear independence is a property of the set, not of the vectors. For example, if in R^3:

a1 = (1,0,0), a2 = (0,1,0), and a3=(1,1,0),

then {a1,a3} is a linearly independent set of vectors and {a1,a2,a3} is not. But you can't say a1, a2, and a3 are linearly independent, as that implies it is a property of each vector, which then implies that a1 and a3 are linearly independent. I understand it's a matter of using fewer words, and that no one ever gets confused when someone says a1 and a2 are linearly independent vectors. But at least let's try to get it right.