THE SPACE R AND FUNCTIONS ON R. The space r. The space r. Functions from r to r. PRODUCTS OF VECTORS AND ELEMENTARY TOPOLOGY R. The dot product in r. Products in r. The dot product in r. VECTOR SPACES. The definition of vector space. Properties of vector spaces and subspaces. The span of a set. LINEAR TRANSFORMATIONS AND MATRICES. Linear transformations. Matrices. Lines, planes, and hyperplanes. DIFFERENTIAL CALCULUS. The derivative of a function. Directional and partial derivatives. The jacobian matrix. INNER PRODUCT SPACES. Inner product spaces. Orthogonal transformations. Quadratic forms. DETERMINANTS AND SYSTEMS OF EQUATIONS. Definition of the determinant. Properties of the determinant. Systems of linear equations. MAPPING THEOREMS AND EXTREME VALUES. Inverse function theorem. Implicit function theorem. Higher order derivatives. INTEGRAL CALCULUS. Iterated integrals. Multiple integals. Curvilinear coordinates. NORMAL FORMS. Invariant subspaces and symmetric transformations. Diagonalization of symmetric transformations. Jordan normal form.

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