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A Basic Introduction to Linear Algebra

First, a stupid review of integers, just to emphasize the points where we will later get confused. Suppose you have three integers, a, b, and c. abc = acb = bac = bca = cab = cba. Order is completely unimportant. Integers commute. When you multiply two integers, you get an integer. Integers are closed under multiplication. If you have 3 pool balls and you speed up to .87c, you still have three pool balls. If you rotate 90 degrees, you still have three pool balls. If you drive from LA to New York, you still have three pool balls. These statements are all true whether you take the pool balls with you or leave them behind. All observers agree that there are three pool balls. Integers do not participate in translations, rotations or boosts. That's it for our stupid review of integers.

Vectors do not commute. Matrices do not commute. Order is important. When you multiply two vectors, you get a matrix, not a vector and not an integer. When you multiply two matrices, you get a tensor, not a matrix. Vectors are not closed spaces under multiplication. When you translate, rotate, or boost, the representation of all the vectors and matrices change. Different observers will have different opinions about the wind velocity and direction. One must keep all these things in mind. This is not so easy for the beginner - we're taught the rules of commutation and closure long before we're even taught the words. The concepts of commutation and closure are deeply engrained. Sorry, those concepts now need a subscript in your brain, it's now got to be "Things Commute_(sometimes)." A simple example: Take two steps forward. Rotate 90 degrees to your left. Remember where you wound up. Now, go back to where you started. Rotate 90 degrees to your left. Take two steps forward. You're not in the same place, are you?

Actually, things are even worse than this: we live in a subjectively three-dimensional universe. In three spacial dimensions there are some tricks and shortcuts you can use that you cannot use in any other space, and these tricks and shortcuts are taught as if they are universal. Classical electromagnetism is normally formulated as a three dimensional theory, and these tricks and shortcuts are all through it. However, you may not use these tricks and shortcuts when you're doing quantum field theory, general relativity, or special relativity. Example: if you've already learned a bit about vectors, you've learned the dot product and the cross product. Neither of these are a legitimate example of vector multiplication. So, we'll take a somewhat unusual approach to linear algebra here: we'll learn the more general formulation.