(112-1) Introduction to Linear Algebra (1)

Time : Wed 3,4 and Fri 3,4
Credits : 4
Textbook : Linear Algebra, 4th Edition, by  Stephen H. Friedberg, Arnold J. Insel, Lawrence E. Spence 

Course Description 課程簡介

Many problems in science are often resolved by firstly linearizing them. For instance, in calculus, differentiation can be seen as a process of locally approximating functions with linear functions. This process of linearization often sheds new light into the original problem. Linear algebra, the branch of mathematics that systematically deals with linearized functions, is now considered the fundamental language in almost all quantitative scientific studies.

The objective of this course is to introduce the fundamental ideas of linear algebra. In the first semester, our focus will be on studying vector spaces and the linear maps defined between them. Although these concepts are defined in abstract and general terms, we will approach them in a concrete and computable manner using matrices. By the end of the first semester, we will discover that every linear map on a finite dimensional vector space can be represented by an upper triangular matrix, and in some cases, even better, by a diagonal matrix, depending on the choice of a `basis’.

This course primarily focuses on pure mathematics. Theorems will be rigorously proven in their most general form, but key concepts and results will be illustrated through concrete examples from various disciplines.

Provisional Syllabus 預定每週教學進度

1. Fields and Vector spaces, Subspaces

2. Linear independence, Spanning, Basis and Dimension

3. Direct sums and complementary subspaces

4. Linear maps (I) : kernel and image, rank and nullity theorem, isomorphism of vector spaces

5. Linear maps (II) : representation by matrices, transition matrices

6. Linear maps (III) : Gaussian elimination, rank of a matrix

7. Linear maps (IV) : inverse of a linear map, row reductions and elementary matrices, inverse of a matrix

8. Midterm Exam Week

9. Determinants (I) : Definition

10. Determinants (II) : Properties 

11. Similar matrices and their properties

12. Eigenvalues, eigenspaces and eigenvectors

13. Diagonalizablity (I) Algebraic and geometric multiplicities 

14. Diagonalizability (II) Cayley-Hamilton Theorem, Minimal polynomials

15. Applications to difference and differential equations

16. Final Exam Week