Course Introduction

• Prerequisite：high school mathematics
• Recommended for： freshmen
• Introduction：

This course provides a first introduction into the origin, theory and application of Linear Algebra.

• Syllabus：

1. Matrices and Linear System
   Matrices and their algebra, solving systems of linear equations, inverse of square matrices, homogeneous linear systems, computation of inverse matrix, Gaussian elimination, existence and uniqueness of solutions, reduced row ech-elon form, row rank and column rank, nullity
2. Vector Spaces
   Euclidean spaces, general vector spaces, linear subspaces, independence, span, basis and dimension, coordinatization of vectors
3. Linear Transformation
   Invariant subspace, kernel, range, rank-nullity theorem, matrix representation and change of basis, similarity
4. Determinant
   Areas, volumes, cross product, properties of determinant, computation of determinant, Cramer's rule
5. Eigenvalues and Eigenvectors
   Characteristic polynomials, algebraic and geometric multiplicities, eigenspaces, diagonalizations and triangularization, invariant subspaces
6. Inner Product Space
   Inner product, norm, adjoint of a matrix, bilinear forms*
7. Orthogonality and Projections
   Orthogonal matrices, Gram-Schmidt process, mutually perpendicular vectors and subspaces, orthogonal projection.
8. Quadratic Forms
   Diagonalization of quadratic forms, positive and negative definite quadratic forms, applications to extrema
9. Special Matrices
   Tridiagonal, symmetric, projection, unitary, Hermitian, normal, nilpotent matrices and their properties
10. Complex Scalars and Vector Spaces
    Matrices and vector spaces over complex scalars, complex Euclidean inner product, complex subspaces*, Jordan canonical forms*, minimal polynomial*
11. Applications*
    Markov chain, Fibonacci sequence, solving system of ODE, method of least squares, etc.
12. Additional topics as time permits
    Direct sum, direct sum decomposition of vector spaces, direct sum decomposition of linear operators, dual spaces

• Reference：

1. Linear Algebra, by J. B. Fraleigh and R. A. Beaurgard
2. Linear Algebra, by S. H. Friedberg, A. J. Insel, and L. E. Spence
3. Linear Algebra, by K. Hoffman and R. Kunze
4. Linear Algebra and its Applications, by G. Strang
5. Differential Equations, Dynamical Systems, and Linear Algebra, by M. Hirsch and S. Smale