I've been asked by an high school student what are the $5$ major theorems in Lang's Linear Algebra (and therefore, by extension, in an undergraduate linear algebra course). Firstly, I bluntly said that the question holds a great deal of subjectivity and that I couldn't answer it.
However, coming to think of it, I realized that the question is indeed somewhat well-posed. Indeed, it could be natural to ask which of the theorems presented in a course are most useful for the prosecution of the study; and it requires some maturity to pick the key ideas that are most relevant for a subject as a whole.
So, I would like to ask what are the $5$ theorems that best epitomize an undergraduate linear algebra course and communicate the key ideas of the subject.