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Module Objective: To introduce students to linear equations, determinants, vectors, complex numbers and matrices.
Overview of coordinate geometry in 1, 2 and 3 dimensions. Describing lines and planes by means of linear equations. Systems of linear equations and augmented matrices. Row echelon form matrices and back substitution. Elementary row operations and Gaussian elimination. Bound and free variables. Matrices and rules of matrix arithmetic. Noncommutativity of matrix product. Determinants of 2 x 2 and 3 x 3 matrices. Determinant of a product. Effect of elementary row operations on determinants. Determinants as areas and volumes. Matrix inverses, in particular 2 x 2 inverses. Overview of vectors in 1, 2 and 3 dimensions. Position and displacement vectors. Triangle and parallelogram laws. Vector arithmetic, norm, distance, dot product and angle, orthogonal projection. The triangle and Cauchy-Schwartz inequalities. The cross product and triple product in 3-dimensions. The normal vector of a plane and the vector equation of a line. Complex numbers and the plane. Other topics, such as the eigenvalues of 2 x 2 matrices, roots of unity, geometric problems, quadratic forms, linear maps via 2 x 2 matrices.