What ideas have you had so far?
Good approach - to find the eigenvalues and eigenvectors. I think you may have skipped too many steps in taking your determinant. I get
This is not the same thing as what you got, if you multiply it all out (sometimes it pays not to do that!). So, what are the eigenvalues?
Yes, you've correctly identified the diagonal matrix such that there is an invertible matrix such that
The reason you need to diagonalize is because the solution to the system , as you've hinted at, is the following:
So you have to compute this matrix exponential and the best and easiest way to do that is to diagonalize Why? Because it's ridiculously easy to compute arbitrary powers of a diagonal matrix (just raise the elements on the diagonal to the desired power!). Also,
In general, you have
Then you just invoke the Taylor series expansion of the exponential, and you find that
The RHS there is easy to compute: is just the matrix with the elements That is, you just exponentiate the elements on the main diagonal.
So I would say that diagonalization, when it can be done, is at the heart of solving linear systems of ODE's with constant coefficients.
So, what remains to be done is this: construct your invertible matrix and then compute , and then compute and you're done.
Make sense?
Well, why not compute the derivative, and see if it satisfies the original system? Here's one of the really great things about differential equations: as long as you're decent at differentiation (which is quite straight-forward, really), you can check your own answers very easily.
What do you get?
True, but you don't know ahead of time if you made any mistakes or not. I recommend computing the LHS (the derivatives), and then computing the RHS and show they are equal.well, if i didnt make any mistakes the solution should be correct!
As for the linear subspaces, could you please state your question a little more carefully? Perhaps rephrase it a bit? I'm not following what you're asking, though I have an inkling.