Linear Algebra (MA283) : Course Outline for 2008-2009 (Semester II)
The course should give a solid understanding of basic notions in linear
algebra such as
- vector space,
- matrix representation of a linear map,
- eigenvectors and matrix diagonalization,
- Gram-Schmidt and orthogonal bases.
The course should develop an ability to
- extract abstract mathematics from specific examples
- read mathematics books
- write clear and precise mathematical statements
- Linear functions:
Definition of a linear function and four examples. (1) Derivatives.
(2) Polynomial multiplication (BCH codes). (3) Symmetries
(molecular symmetry). (4) Projections (computer graphics).
- Representations of linear functions: Description of linear functions by a
"small" set of values. Application to the above four examples.
- Bases and Vector spaces: Extraction of precise definitions from
the preceding examples.
- Eigenvalues and matrix
diagonalization: Observation that
nice bases give
rise to nice matrix representations. Hamilton-Cayley theorem.
- Kernels and images of linear maps:
The formula Dim(Ker(f)) + Dim(Image(f)) = Dim(Source(f)) . Row space,
nullspace and rank of a matrix.
- Linear equations and row operations:
Including inversion of a square matrix.
- Inner product spaces: Including Fourier series.
Linear Algebra and its Applications by Gilbert Strang (4th Edition, paperback).
Professor Strang's MIT lectures based on this book can be found on YouTube and are a good complement to my lecures. (Google: "strang" + "linear algebra" +YouTube")
You might also find the wikipedia web
pages on linear algebra quite useful.