The definition of the tensor product and existence/uniqueness/associativity properties.
I know, this is perhaps not a single theorem but in my eyes one of the most useful "elementary" concepts. Personally, I had two semesters of linear algebra without mentioning the tensor product. And from this I suffered for a long time during my further studies. Now it is my first homework/exercise for students in my lectures (e.g. diff geo).
If the student is really clever, one can even do something like the tensor algebra in these 30 min.
