Definitions. Definitions are the foundation on which mathematics is built. There are many definitions in this course, more than in any course you have previously had. If you don't know the definitions forwards, backwards, and sideways, you will certainly fail this course. Start learning them the first day of class. Build your own list of definitions. Review them regularly. One reason students find this course so difficult is that they don't make a real effort to understand the definitions. Or, they leave that affort too late. You must begin the first week of the course and constantly monitor yourself. You can monitor your progress by being honest with yourself when I write things on the board. If you don't understand a word I use, ask. There will be others in the class who don't understand it either. By asking you do everyone a favor. Even many of those who think they understand a definition don't.
The document definitions consists of my response to an email from a student who is asking whether he has the correct definitions for some of the terms that arise in Linear Algebra. Mostly the student is close but not close enough---all it takes to sink a boat is one hole, and the same holds for definitions: either it is correct or it is wrong. I always ask for a lot of definitions in my exams because they are so important. If you don't know the definitions you will surely fail this course. Here is another note about definitions and comments on various errors made by students.