28.15. Linear Transformations and Differential Equations

Try these first

1. 1.

   What is the matrix form of a first-order ODE?

   Hint

   Recall how we represent systems in matrix notation.

2. 2.

   What does e^(At) represent?

   Hint

   Think about how we use matrix exponentiation in differential equations.

3. 3.

   What representation can be used to express a system of ODEs?

   • dx/dt = Ay
   • dx/dt = Ax
   • dx/dt = Bz

   Hint

   Think about how we express linear transformations.

4. 4.

   True or False: The solution x(t) = e^(At)x(0) uses the inverse of matrix A.

   • True
   • False

   Hint

   Recall what e^(At) signifies in solving ODEs.

5. 5.

   Consider a 2x2 matrix A with eigenvalues λ1=2 and λ2=-1. Analyze the long-term behavior of a system described by dx/dt = Ax.

   Hint

   Use the properties of eigenvalues to predict system behavior.

6. 6.

   Given a matrix A that you can diagonalize, find the matrix exponential e^(At) when t=1. Start by finding its eigenvalues and eigenvectors.

   Hint

   Remember the diagonalization steps we discussed.