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The University of Texas at Austin
Department of Electrical and Computer Engineering
EE362K: Introduction to Automatic Control—Fall 2009
Problem Set One
C. Caramanis
Due: Wednesday, September 9, 2009.
This problem set is intended to get us started thinking about diFerential equations and their
solution, as well as properties of the solution. In addition, it will give a little practice for some
basic linear algebra.
1. Exercise 1.2 from the book.
2. Using the Taylor expansion for sin, cos, and for the exponential, show (i.e., derive the rela
tionship, do not just quote the result) that
e
iαt
=cos(
αt
)+
i
sin(
αt
)
.
Then based on this, conclude that for
a
∈
C
, the magnitude of
x
(
t
)=
e
at
,
depends on the real part of
a
, and not on the imaginary part of
a
.
3. Consider the threebythree matrix
13 4
20
−
1
11 2
(a) Show that if vectors
v
1
and
v
2
both satisfy
Av
1
=0and
Av
2
= 0, then for any real
numbers
α
and
β
, the vector
v
=
αv
1
+
βv
2
also satis±es:
Av
=0
.
(b) Compute the set of vectors
v
such that
Av
.
(c) Computer the determinant of the matrix
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 Fall '08
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