18.03 Lecture Notes - Lecture 32: Linear Algebra, Companion Matrix, Phase Space
Document Summary
Describe systems of odes using linear algebra notation. Find a companion matrix expressing an ode of degree 2 or higher as a system of linear odes. Express the basis for homogeneous solutions in terms of eigenvalues and eigenvectors of associated matrix equation. We are going to use our new linear algebra knowledge to start looking at systems of des. All of the important theorems about solutions to a single ode generalize to systems of odes. Existence and uniqueness theorem for a linear system of odes. This system can be written in matrix form, x" = a(t)x by de ning involving two unknown functions, x(t) and y(t) is a rst order homogeneous linear. Let a(t) be a matrix-valued function and let g(t) be a vector-valued function, both continuous on an open interval i. Let a be within i and let b be a vector.
