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The symbols are used to indicate the state of a spin 1/2 particle. The arrows indicate spin up (m = 1/2) or spin down (m = -1/2). The representation of these states in terms of column vectors is given in the middle of the page. They are operated on by the various 2x2 matrices called the Pauli matries.
Multivariable calculus is not much help with this part of quantum physics. Basic matrix algebra is necessary. This really requires a lot of background knowledge that should be provided by your instructor. I don't know how you can be expected to work problems without the necessary information.
You can skip the first hour of lecture 1, but starting at around 1:00 he begins talking about math that we have been discussing. Lecture 2 starts the discussion of electron spin and the math used to describe it. Lecture 2 is very good. It should answer many of the questions you had at the start of this thread.
Lecture 3 covers the postulates of quantum mechanics, some general theorems, and the mathematics of spin 1/2 particles in much detail. There is much work with the Pauli matrices although he doesn't use the term. All relates to the original question on this thread.