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1 By the Diagonalization Theorem and Theorem 7 of Ch 5 (dimension of each eigenspace equals the
multiplicity of the corresponding eigenvalue), is diagonalizable as
with 1
0
0 0
2
0 0
0 and
2 13
01
20 1
0
1 2. [10 pts] Consider a dynamical system described by the difference equation x
x
0,
3/4
1/4
1
3
where
. The vectors v
and v
are eigenvectors of corresponding
1/4 17/12
3
1
2
.
to eigenvalues 2/3 and 3/2, respectively. Let x be a solution of the difference equation for x
2
Find a formula for x that does not directly involve the matrix , and describe in words what happens to
∞.
x as , s.t. x Because 1, Find v1 v.
1, goes to ∞ for large . v 312
~
132
Row reduce [v1 v x 0.4 v1 goes to 0 for large k, and because So for large k, x So x v for solution x v1 0.8 132
1
~
312
0 v , and for large k, x is a multiple of v , and the growth rate is 3/2. 3
10 0.8 2
132
1
~
~
8
0 1 .8
0 0
1 .4
.8 1
. That is, as k goes to infinity, x
3 3. [8 pts] Consider p 1 3 ,p 2 3 ,p 1 2. (i) Use coordinate vectors to show that these polynomials form a basis B for
(ii)...
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This note was uploaded on 01/17/2014 for the course ESE 309 taught by Professor Staff during the Fall '08 term at Washington University in St. Louis.
 Fall '08
 Staff

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