Learning Outcomes:
Students are expected to achieve the following competence:
• Giving algebraic and geometrical representations for vectors
• Performing arithmetic with vectors;
• Using vectors to solve classical geometric problems;
• Manipulating dot and cross products, and using them to solve geometric problems;
• Solving systems of linear equations in terms of Gaussian elimination and Cramer’s rule;
• Performing matrix arithmetic;
• Calculating inverse of a matrix and a determinant;
• Solving eigenvalue problem for a matrix;
• Diagonalizing a matrix;
• Solving a quadratic form problem in terms of matrix diagonalization;
• Being able to identifying the rank of a matrix;
• Giving the space spanned by a number of vectors.
Indicative Module Content:
Teaching Plan
Week 1: Appendix: Revisit to high school mathematics: Set theory, summation and product notations, mathematical induction.
Definitions & properties of vectors.
Week 2: Definitions & properties of vectors, scalar multiplication, linear independence. Tutorial 1.
Week 3: Dot product (algebraic & geometric definitions), projection of vectors, two important inequalities, applications.
Week 4: Cross product (algebraic & geometric definitions), applications. Tutorial 2.
Week 5: Applications of vectors: equations of lines and planes, scalar triple product.
Week 6: Matrices and systems of linear equations, Gaussian elimination. Tutorial 3.
Week 7: Row echelon form, reduced echelon form, homogeneous systems.
Week 8: Revision for mid-term exam. Tutorial 4.
Week 9: Matrix algebra, elementary matrices, partitioned matrices, LU decomposition.
Week10: Determinant of a matrix, properties of determinants. Tutorial 5.
Week11: Adjoint matrix, Cramer’s rule.
Week12: Definitions of eigenvalue and eigenvector of a matrix. Tutorial 6.
Week13: Diagonalization of matrix.
Week14: Quadratic forms, positive and negative definite matrices, rank of matrix. Tutorial 7.
Week15: Vector spaces and subspaces spanned by vectors.
Week16: Revision for final exam.
Remarks:
1. In each odd week there are two lectures; in each even week there is one lecture and one tutorial.
2. The chapter of vector spaces is only a brief review of the theory. Interested students are encouraged to read reference textbooks, and wait for future advanced courses on algebra theory.