**Background and Goals:**

Differential equations arise in many areas of science and engineering to model continuous change, in particular to model time evolution of a system. Linear algebra is the study of linear systems of equations, linear functions, matrices and vector spaces. In math 320, we shall study these subjects together for three reasons: (1) The viewpoint of linear algebra is immensely helpful in uncovering the order underlying the topic of differential equations; it helps us understand the ``why'' and not just the ``how'' of our calculations, (2) Linear algebra is essential to the theory of differential equations, and (3) linear algebra is crucial to the computer approximations which are often the only way to solve the most challenging ordinary and partial differential equations.

**Alternatives:**Math 319 focuses on differential equations and avoids using linear algebra concepts. Math 340 is a standard first course in linear algebra. Combining both topics in a single course, as in Math 320, is intellectually sensible but demanding since both differential equations and linear algebra are covered in a single course. Math 320 focuses on linear systems of differential equations such as those arising from modeling of electrical networks and coupled mass-spring systems.

**Subsequent Courses:**Math 320, 321, 322 are a core mathematics sequence for the physical sciences. Math 513 (numerical linear algebra) and 514 (numerical analysis) investigate the numerical methods and algorithms needed to construct approximations to calculus, differential equations and linear algebra problems.

- First-Order ODEs
- Mathematical Modeling and Numerical Methods
- Linear Systems and Matrices
- Vector Spaces
- Higher-Order Linear ODEs
- Eigenvalues and Eigenvectors
- Linear Systems of ODEs
- Matrix Exponential Methods (to solve non-homogeneous systems)
- Nonlinear Systems