 Prerequisite：high school mathematics
 Recommended for： freshmen
 Introduction：
This course provides a first introduction into the origin, theory and application of Linear Algebra.
 Matrices and Linear System<ol>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 echelon form, row rank and column rank, nullity</ol>

Vector Spaces<ol>Euclidean spaces, general vector spaces, linear subspaces, independence, span, basis and dimension, coordinatization of vectors</ol>

Linear Transformation<ol>Invariant subspace, kernel, range, ranknullity theorem, matrix representation and change of basis, similarity</ol>

Determinant<ol>Areas, volumes, cross product, properties of determinant, computation of determinant, Cramer's rule</ol>

Eigenvalues and Eigenvectors<ol>Characteristic polynomials, algebraic and geometric multiplicities, eigenspaces, diagonalizations and triangularization, invariant subspaces</ol>

Inner Product Space<ol>Inner product, norm, adjoint of a matrix, bilinear forms*</ol>

Orthogonality and Projections<ol>Orthogonal matrices, GramSchmidt process, mutually perpendicular vectors and subspaces, orthogonal projection.</ol>

Quadratic Forms<ol>Diagonalization of quadratic forms, positive and negative definite quadratic forms, applications to extrema</ol>

Special Matrices<ol>Tridiagonal, symmetric, projection, unitary, Hermitian, normal, nilpotent matrices and their properties</ol>

Complex Scalars and Vector Spaces<ol>Matrices and vector spaces over complex scalars, complex Euclidean inner product, complex subspaces*, Jordan canonical forms*, minimal polynomial*</ol>

Applications*<ol>Markov chain, Fibonacci sequence, solving system of ODE, method of least squares, etc.</ol>

Additional topics as time permits<ol>Direct sum, direct sum decomposition of vector spaces, direct sum decomposition of linear operators, dual spaces</ol>
 Linear Algebra, by J. B. Fraleigh and R. A. Beaurgard

Linear Algebra, by S. H. Friedberg, A. J. Insel, and L. E. Spence

Linear Algebra, by K. Hoffman and R. Kunze

Linear Algebra and its Applications, by G. Strang

Differential Equations, Dynamical Systems, and Linear Algebra, by M. Hirsch and S. Smale
