Tips for Learning Geometry

Video Transcript

"When you're trying to learn a new concept in Geometry or a new subject, sub-subject of Geometry, there's two things to keep in mind. First the most important thing is going to be the definitions. Our focus should be on understanding the definitions. To do so, you should create examples. Each time you read a new definition, start by creating a simple of an example as you can, and then create more of varying complexity. Draw pictures of them as you go. Just as important for solving problems but less important for the understanding of a subject are the theorems that you'll find in the book. For these as you go along, make a handy list of the important ones for reference when you get to a problem solving stage. Here's an example definition, the multiplication and addition here is vector multiplication and addition by scalers. Now the coefficients of this linear combination have to sum up to one. So here's a simple example. Suppose there are only two points, x one and x two. The example should be concrete. I've actually got this x one chosen to be two comma four. And x two chosen to be five comma one. Now a linear combination of those would be a number A times x one plus another number times x two. But if those linear coefficients have to sum to one, then the second number is just going to be the difference between one and the first. So I call it one minus A. Now in this example after a little bit of thought, I can see that the set of all such points is actually the line segment connecting x one and x two. Now a more complicated example. Say we've got five points. I just randomly pick them on the plane. Now after trying to trying linear coefficients, it turns out that the convex whole of theses points consists of not only the lines connecting them, but all of the area inside those outermost lines. In fact, a nice description of the convex whole in more easier terms is to describe taking a piece of wood, then sticking in a thumbtack for each of the points, and then taking a rubber band around the entire group. Now the convex whole is the area included inside of that rubber band boundary."