In addition to learning the customary computational skills in calculus and
linear algebra, the student will be exposed to mathematics as
mathematicians view it. Students will learn to write proofs and think more
rigorously about mathematics, and will come to grips with challenging
concepts and problems. In that regard, this course is an excellent preparation
for students considering a career in law, medicine, or the sciences.

1. Vectors, dot product, subspaces, linear transformations and matrices;
introduction to determinants and cross product.
2. Scalar- and vector-valued functions, topology of euclidean space, limits
and continuity.
3. Differentiation: directional derivatives, differentiability, differentiation rules,
the gradient, curves, and higher-order partial derivatives.
4. Implicit and explicit solutions of linear systems: Gaussian elimination,
elementary matrices and inverses, linear independence, basis, and dimension, the four
fundamental subspaces associated to a matrix.
5. Extremum problems: compactness and the maximum value theorem,
maximum/minimum problems, the second derivative test, Lagrange 
multipliers (with an introduction to eigenvalues), projections and least-
squares solution of inconsistent systems.