MATH416
MATH416 (Abstract Linear Algebra) is a 3-credit-hour class is a 400 level rigorous treatment of linear algebra that may be used to satisfy the linear algebra requirement for EE and CE majors.
Content Covered
- Linear equations, linear systems
- Matrix operations, gaussian elimination
- Vector spaces, subspaces, inner product spaces
- Basis and dimension
- Linear transformations, adjoints
- Matrix representation of a linear transformation
- Eigenvalues, eigenvectors, diagonalizability and its criterion
- Cayley Hamilton Theorem
- Inner products and norms
- Orthogonality, orthogonal projections, orthogonal complements
- Gram-Schmidt method
- Jordan Canonical Forms
MATH416 is likely the first linear algebra course that will give you a basis in linear algebra. You start with the notion of vector spaces and subspaces, then moving to matrices and linear systems. After this you begin to focus on bases and dimension, which allows you to consider things such as linear transformations, diagonalizability, linear independence, inner products, and more. The class ends with Jordan Canonical Forms, a powerful tool to "almost" diagonalize any matrix over a valid field.
Prerequisites
MATH241 is the only official prerequisite for this course, as the math department does not enforce strict prerequisites.
When to Take It
Students should take this course after recieving credit in MATH347 or MATH314 as it will allow you to have a greater understanding of the proof techniques used in the course.
Course Structure
Like a typical math class, this course has just 2/3 lectures during the week.
Homeworks tend to vary by instructor, and its not very standardized. You should expect ~1 homework per week, likely submitted on canvas or gradescope.
Midterms also vary by instructor, but you should expect at least 2 midterms and a final. These will be varying in length but will be proctored using pen and paper.
Instructors
The instructors for this course vary based on availability and student demand. Professor Albin often teaches this class and is regarded as one of the best professors to take it with.
Course Tips
- The midterms can take a lot of time depending on the content, and will often feature shortcuts for students who can recognize the chance to apply it. Make sure you understand the content well.
- Do your homework problems in their entireity with minimal help from AI as it will train you to get in the mindset of recognizing familiar concepts.
- Make sure you focus on understanding the basics early on as it is the foundation for everything else you learn later on in the class.
- 3Blue1Brown has great videos online for learning the intuition behind the topics that you are seeing in class in his "The essence of linear algebra" playlist.
- Use office hours as a tool for learning, it is the best resource you have available.
Life After
After taking MATH416, ECE students are required to take ECE313, which delves into probability theory. MATH416 will also give you a strong foundation in linear algebra that will prepare you for AI/ML classes such as ECE448 and ECE449
Infamous Topics
There are a lot of proofs that you need to understand and know at any given time in the class. Among the most rigorous are the diagonalizability criterion and the exixtence of Jordan Canonical Forms. Many students also find determinants and proving their properties to be difficult, as they have only clunky definitions that are hard to manipulate/work with. Lastly, when the class expands the field to complex numbers, there are additional caveats you must remember and can be a lot to keep track of.
Resources
As mentioned before, 3Blue1Brown has an amazing video series on linear algebra, in which he includes well-made visualizations of certain concepts learned in the course.