**Course** Description: | Vector algebra and geometry, fundamental concepts of linear
algebra, linear transformations, differential calculus of
functions of several variables, solutions of linear systems and
linear independence, extremum problems and projections. This
course and its sequel give an integrated and more proof-oriented
treatment of the material in Multivariable Calculus and
Introduction to Linear Algebra. |

**Course Objectives:** | 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. |

**Topical Outline:** | 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. |