Tools and concepts from linear algebra are key to many branches of mathematics, including functional analysis, graph theory, abstract algebra, representation theory, numerical analysis, differential equations, and much more. Basic linear algebra tools -- especially eigenvalues and eigenvectors --aren crucial for the study of dynamical systems. Linear algebra also has fascinating applications in and of itself. One famous one is the PageRank algorithm used by Google to rank search results. There is a wonderful, well-explained article about this algorithm. Another wonderful example comes from facial recognition techniques that use eigenfaces, which are essentially basis elements that span the space of possible appearances of the human face.
By the end of this lesson, you should be able to:
- Perform matrix/vector arithmetic.
- Define and calculate eigenvalues and eigenvectors.
- Relate the eigenvalues of a matrix to its trace and determinant.
- Diagonalize a matrix.
Before class, please:
- Read Olver and Shakiban excerpt 1 (all) and excerpt 2 (sections 8.2 - 8.3).
- Watch whichever of these you need for adequate review: a quick intro to vectors, a quick geometric review of some vector concepts, a reminder of some basic matrix operations, this introduction to the idea of eigenvalues and eigenvectors, this introduction to calculating eigenvalues and eigenvectors, this review of diagonalization.
In class, we will:
After class, please:
- Do post-class problems.