What Should I Focus on When Preparing for Graduate-Level Algebra I?

[futurebird]

I'm getting ready to take grad level Algebra I. My adviser gave me a book to read over the summer. He said "read the first 50 pages." I audited undergrad Algebra I, but I'm wondering what concepts should I focus on for my self-study preparation?

1. What are the hardest things to understand about Algebra I. I'm talking about concepts like "compactness" in Analysis (that took a few days for me to really grasp.)

2. What are the most important theorems? You know the big ones you really need to know.

3. What is "cool" or "fun" about this subject? What can I look forward to?

Are there any internet resources or books you would recommend?

My plan right now is to read the first 50 pages. Memorize the definitions. Do the problems I can find answers for on the web and maybe learn two of the important proofs by heart. (not memorize them, but so I can pretty much do them on my own.) I hate feeling overwhelmed, and since my undergrad degree was in Drama (yes Drama) I often find I need to do a lot of back-tracking to fill in my gaps. (but this is getting better after my first year.) So, I want to be extra prepared and lessen the misery. If possible.

 

[mathwonk]

go to my website http://www.math.uga.edu/~roy/

and download the 80006a notes: course outline.

if interested download the rest of the notes.

or my more detailed 843-4-5 notes.

these were actual notes from a grad algebra I course taught at various times over 1-2 semesters.

 

[Mene]

I can understand your desire to be extra prepared and lessen the potential difficulties in understanding Algebra I at the graduate level. Here are my responses to your questions:

1. In my opinion, some of the hardest concepts to understand in Algebra I are abstract algebra and linear algebra. These topics involve understanding structures and operations on sets of numbers or other mathematical objects, which can be challenging for some individuals. Another concept that may be difficult to grasp is the concept of variables and their manipulation in equations, as this requires a strong understanding of the underlying principles of algebra.

2. The most important theorems in Algebra I vary depending on the specific course and curriculum, but some of the key ones include the fundamental theorem of algebra, the quadratic formula, and the properties of exponents and logarithms. These theorems serve as the basis for many other concepts and applications in algebra.

3. The "cool" and "fun" aspect of Algebra I is that it is a fundamental and powerful tool for solving real-world problems and understanding the underlying structures of mathematics. It allows us to make connections between different mathematical concepts and apply them in various fields such as physics, engineering, and economics. Additionally, as you progress in your studies, you will see how algebra is the foundation for more advanced mathematical concepts and theories.

Some internet resources and books that I would recommend for self-study in Algebra I are Khan Academy, MathIsFun, and "Algebra: A Complete Introduction" by Hugh Neill and Rachel Hilliam. These resources provide a variety of practice problems and explanations that can help you solidify your understanding of the concepts.

In terms of your plan, I think it is a good idea to start by reading the first 50 pages and focusing on memorizing the definitions. This will give you a strong foundation for the rest of your studies. Additionally, practicing problems and learning important proofs will also help you to better understand the material. Remember to not get too overwhelmed and take breaks when needed. Good luck with your studies!