Brief Summaries of Topics

Summary

Linear Algebra

Linear algebra studies linear transformations and vector spaces, or in another language, matrix multiplication and the vector space Rⁿ. 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 also know why eigenvectors and eigenvalues occur naturally in linear algebra.

Real Analysis

The basic definitions of a limit, continuity, differentiation and integration should be known and understood in terms of ∈'s and δ's. Using this ∈ and δ language, you should be comfortable with the idea of uniform convergence of functions.

Differentiating Vector-Valued Functions

The goal of the Inverse Function Theorem is to show that a differentiable function f : Rⁿ → Rⁿ is locally invertible if and only if the determinant of its derivative (the Jacobian) is non-zero. You should be comfortable with what it means for a vector-valued function to be differentiable, why its derivative must be a linear map (and hence representable as a matrix, the Jacobian) and how to compute the Jacobian.