# Precalculus Resources: Spring 2017 Midterm I Review (Section 0.1 – 2.2)

I have gathered quizzes that relate directly to the upcoming midterm, as well as included a sample midterm. Please feel free to use these for practice! If you have trouble with the material, visit this page for additional resources. Don’t forget to grab a free version of wolfram alpha pro to help you if you’re enrolled at San Francisco State University!

# Notes: Modern Algebra II

I took Modern Algebra II with Professor Matthias Beck in Spring 2016. These comprehensive notes were compiled using lecture notes and the textbook, David S. Dummit & Richard M. Foote, Abstract Algebra (3rd edition), Wiley 2004. [errata]

Featured Image: Icosahedron and dodecahedron Duality
Credit: Images from Algebra: Abstract and Concrete by Frederick M. Goodman

Disclaimer: my notes are meant to be a toolbox while doing proofs and studying/practicing the course in general. There may contain typos or mistakes. Please feel free to let me know if you find any errors!

Topics Covered:

• Review of basic properties of groups and rings and their quotient structures and homomorphisms,
• group actions,
• Sylow’s theorems,
• principal ideal domains,
• unique factorization,
• Euclidean domains,
• polynomial rings,
• modules,
• field extensions,
• primitive roots,
• finite fields.

# San Francisco State University Math Resources: Mathematica & Wolfram Alpha Pro

SF State has a site license for Mathematica. Under this site license, Mathematica can be installed without further cost on any computer owned by San Francisco State University, by SF State faculty, or by SF State students. This software works on Windows, Macintosh, and Unix.

Under the terms of the site license negotiated by CSU, free WolframAlpha Pro accounts are also now available to our faculty, students, and staff.

If you are an SFSU student, click here for a free WolframAlpha Pro account.

### Mathematica

1. Create an account (New users only):
1. Go to user.wolfram.com and click “Create Account”
2. Fill out form using a @sfsu.edu email, and click “Create Wolfram ID”
1. Fill out this form to request an Activation Key
4. Run the installer on your machine, and enter Activation Key at prompt

For both Mathematica and WolframAlpha Pro requests on the links above, it is important that only sfsu.edu email addresses be used. Requests from other email addresses will be rejected.

# Precalculus Resources: Intro & Review of Functions

In the first couple of chapters, Dr. Axler covers the properties of real numbers, and gives a throughout exploration of functions in his textbook for Math 199 (Precalculus at SFSU).

### Flashcards

You can take the file to a print shop to have directly printed and cut for you, or print it double sided on regular paper and cut + paste onto index cards. Get creative!

More resources to come!

# Notes: Number Theory and Modern Algebra I

I took Modern Algebra I with Professor Matthias Beck in Fall 2015. These comprehensive notes were compiled using lecture notes and the textbooks,

Disclaimer: my notes are meant to be a toolbox while doing proofs and studying/practicing the course in general. There may contain typos or mistakes. Please feel free to let me know if you find any errors!

Topics Covered:

• Integers & the Euclidean algorithm
• Complex numbers, roots of unity & Cardano’s formula
• Modular arithmetic & commutative rings
• Polynomials, power series & integral domains
• Permutations & groups

Featured Image: Dodecahedron-Icosahedron Duality
Credit: Images from Algebra: Abstract and Concrete by Frederick M. Goodman

# Notes: Mathematics of Optimization

I took this course with Professor Serkan Hosten in Fall 2016. These comprehensive notes were compiled using lecture notes and the textbook, Optimization Models by Giuseppe Calaore and Laurent El Ghaoui, Cambridge University Press.

Featured Image: Analysis explanation of why when optimization a linear objective function over a convex shape will lead to a optimal solution on the boundary of the feasible region.
Image Credit: Figure 08-19 in Optimization Models by Giuseppe Calaore and Laurent El Ghaoui, Cambridge University Press.

Disclaimer: my notes are meant to be a toolbox while doing proofs and studying/practicing the course in general. There may contain typos or mistakes. Please feel free to let me know if you find any errors!

Syllabus:

1. Optimization Models: modeling optimization problems as linear, quadratic, and semidenite programs,
2. Symmetric Matrices: using spectral decomposition of symmetric matrices for positive denite matrices and their role in optimization
3. Singular Value Decomposition: computing SVDs in the context of linear equations and optimization,
4. Least Squares: solving systems of linear equations and least squares problems,
5. Convexity: identifying key properties of convex sets and convex functions for optimization,
6. Optimality Conditions and Duality: developing criteria to identify optimal solutions and using dual problem, in particular, in the context of linear models, Linear and Quadratic Models: employing the geometry of linear and quadratic models for solution algorithms,
7. Semidefinite programs: modeling certain convex optimization problems as semidefinite programs

Fun fact: The subtitle for the site, “#teamnosleep” originated from my study group in this class. Homework assignments were intensely hard.