Math 350 - Linear Algebra
Description
Math 350 will revisit a lot of the same ideas and concepts that were introduced in Math 250, but with two key differences. The first is that the ideas are formulated for arbitrary vector spaces (which can be seen as a generalization of the usual Euclidean space ). The other is that this class places a larger emphasis on proof writing than Math 250, with students expected to write properly written proofs as learned in Math 300. Some new topics that are not covered in 250 are also introduced.
Math 350 is one of the courses that can be used to satisfy the math major’s algebra course requirement, with the other one being Math 351 (Introduction to Abstract Algebra). For students wishing to pursue the honors track in mathematics, it should be noted that Math 350 (preferably its honors section Math 350H) is recommended preparation for the honors algebra sequence. The differences in topic coverage between the regular and honors section is typically minimal.
Topics covered
Topics covered in Math 350 typically include:
- Definitions and examples of fields and abstract vector spaces
- Subspaces
- Span, linear independence, basis, and dimension
- Linear transformations, null spaces, ranges
- Matrix representations of linear maps, composition of linear maps (as matrix multiplication), and invertibility of linear maps (as matrix inversion)
- Change of basis/coordinates
- Systems of linear equations
- Determinants
- Eigenvalues, eigenspaces, and diagonalization
- Invariant subspaces and the Cayley–Hamilton theorem
- Generalized eigenspaces and Jordan Canonical form
- Inner product spaces
- Orthogonal/orthonormal bases and Gram-Schmidt
- Orthogonal transformations and matrices
Note that the exact list of topics may vary by instructor and semester. Additional topics may include Zorn’s lemma and infinite dimensional vector spaces, dual spaces, Fourier coefficients/series, minimal polynomial, rational canonical form, normal and self-adjoint operators on inner product spaces, and bilinear and quadratic forms.
Resources
Textbook
The usual textbook for the class is Linear Algebra (5th edition) by Stephen Friedberg, Arnold Insel, and Lawrence Spence.
Additional resources
Another good resource is Linear Algebra Done Right (4th edition) by Sheldon Axler. This book can freely be downloaded from the author’s website at this link: https://linear.axler.net/. Note that this book has a different approach than the one in Friedberg, Insel, and Spence and so the selection and ordering of topics can vary significantly.
A course webpage from Fall 2022, taught by William Y. Chen, may be found here: https://www.williamyunchen.com/f22math350.