Concrete Theory Introduce the basic computational tool. Introduce some basic concepts in concrete and familier setting

**Chapter 1. Systems of linear equations** The analysis of how systems of linear equations are solved leads us to the basic computational tool of this course: row operation and row echelon form. The tool provides explicit answer to the existence and uniqueness questions for systems of linear equations. It also yields some other results that will be extended to other contexts and more abstract settings.

**Chapter 2. Linear transformation and matrix** System of linear equations are considered from the geometrical viewpoint of linear transformations (between euclidean spaces). The existence and uniqueness problems for the equations have natural interpretations (onto, one-to-one)from the new viewpoint. Moreover, linear transformations are equivalents to matrices. The matrix operations, especially the matrix product and inverse matrix, are natrually introduced from the trnasformation viewpoint.

Abstract Theory The core part of linear algebra. Extend concrete theory to abstract setting and introduce more abstract concepts.

**Chapter 3. Vector space** The concepts of vector spaces and subspaces are introduced. Linear transformations are extended to the more abstract setting. Abstract concepts related to the existence (range, column space, span) and uniqueness (kernel, nul space, linear independence) are introduced. Techniques for dealing with abstract linear algebra problems are developed (basis, coordinates), culminating in the statement that abstract problems can always be translated into concrete problems. With the help of the technique, more abstract theories such as dimension, rank, dual and direct sum are established.

Special Topics More advanced part of the course. The topics are theoretically the most important and practically the most useful.

**Chapter 4. Determinant** First the geometric meaning of determinant is discussed. Then the computational technique is developed and propertied are listed. Finally, the definition of determinant is rigorously developed and the properties are rigorously proved.

**Chapter 5. Eigenvalue and eigenvector** The concepts of eigenvalue and eigenvector are introduced geometrically. Then the computational technique and the matrix interpretation is developed. Moreover, the meaning of complex eigenvalues and more properties are discussed. The applications to differential equations and dynamic problems are given. Further advanced theories are also outlined.

**Chapter 6. Inner product** The concept of inner product is introduced. Its applications to the geometrical measurement and duality are discussed. Then orthogonality theory is developed. Finally, the theory of complex inner product is briefly outlined.

**Chapter 7. Symmetric matrix** The theory of symmetric matrices is a beautiful combination of the eigen theory and the inner product theory. It also has important applications to the theory of quadratic forms.