**Brief Summaries of Topics**

**0.1 Linear Algebra**

Linear algebra studies linear transformations and vector spaces, or in another language, matrix multiplication and the vector space **R**^{n}. You should know how to translate between the language of abstract vector spaces and the language of matrices. In particular, given a basis for a vector space, you should know how to represent any linear transformation as a matrix. Further, given two matrices, you should know how to determine if these matrices actually represent the same linear transformation, but under different choices of bases. The key theorem of linear algebra is a statement that gives many equivalent descriptions for when a matrix is invertible. These equivalences should be known cold. You should ...

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