Math 224-01: Differential Equations: Reading Homework 7.6

1. Undriven Linear Systems: Complex Eigenvalues : how does our solution to a system change when eigenvalues are complex? what would we generally rather have?
   
   1. Complex Vectors : what is Euler's formula for complex exponentials e^{(a + ib)t}? how can we write vectors with complex entries?
      
   2. Finding Real-Valued Solutions : what do we first find when we are looking for all the real-valued solutions to a system x' = Ax? what fact about the complex valued solutions do we use?
      
      1. Point : what are the steps that we follow to find real-valued solutions to a system x' = Ax?
         
      2. Point : how, in example 7.6.1, is the transition between the solutions in equation (6) to the real-valued solutions in (7) made?
         
   3. Another Approach : what is the existence and uniqueness theorem for linear systems? what does it add to the fundamental theorem?
      
      1. Point : what is a basic solution set? how can we test for it?