Description
Exercises on differential equations and eAt
Problem 23.1: (6.3 #14.a Introduction to Linear Algebra: Strang) The matrix
in this question is skew-symmetric (AT = −A) :
⎡ ⎤
du
0 c −b u1
� = cu2 − bu3
dt = ⎣ −c 0 a ⎦ u or u2
� = au3 − cu1
b −a 0 u3
� = bu1 − au2.
Find the derivative of ||u(t)||
2 using the definition:
||u(t)||
2 = u1
2 + u2
2 + u3
2.
What does this tell you about the rate of change of the length of u? What
does this tell you about the range of values of u(t)?
Problem 23.2: (6.3 #24.) Write A = 1 1
as SΛS−1. Multiply SeΛt
S−1
0 3
to find the matrix exponential eAt
. Check your work by evaluating eAt and
the derivative of eAt when t = 0.
1
18.06SC Linear Algebra
