Linear Algebra Notes

These are some rather extensive (33 pages in total) review sheets that I made for my linear algebra class in the Fall 2004 semester. The book we were using turned out not to be that helpful for the students, and so I decided to do these review sheets. The class consisted mostly of biology majors. Next time I teach linear algebra, there are number of changes I'll make to these notes. For example, there's an easier way to find the orthogonal complement than the one in the notes, and I'll probably add a section on the different ways of looking at matrix multiplication. Update - December 2005 These have now been revised. I fixed any typos I found and put in illustrations that earlier had been done by hand.

  1. Brief Overview of Row Operations, LU Factorization, Transposes, Vector Subspaces, Nullspace, Solving Ax=b, Rank, Column Space, Linear Independence.

  2. Orthogonal Subspaces, Orthogonal Complement, Projections, Gram-Schmidt.

  3. Basis, Bases for Nullspace, Row Space, and Column Space, Determinants, Cramer's Rule, Inverses by Cofactors, Eigenvalues and Eigenvectors, Diagonalization.

  4. Symmetric Matrices and Orthogonal Diagonalization, Markov Matrices, Positive Definite Matrices, Quadratic Forms, Similar Matrices, Singular Value Decomposition.

  5. Linear Transformations, Matrix of a Linear Transformation, Change of Basis, Geometry of Linear Transformations, Compositions and Inverses, Kernal and Range, Approximation of Dominant Eigenvalues and Eigenvectors.