Unformatted text preview: M346 Second Midterm Exam, November 9, 2004
22
.
1. Let A =
13
a) Find the eigenvalues and eigenvectors of A.
b) Write down the most general solution to the diﬀerence equation x(n) =
Ax(n − 1).
c) Find x(n) when x(0) = (7, 1)T .
3
2
2. Let A =
and consider the diﬀerential equation dx = Ax.
dt
−5 −4
a) Find the eigenvalues and eigenvectors of A.
b) How many stable modes are there, and what are they? How many unstable
modes are there and what are they? How many neutrally stable modes are
there and what are they? What is the dominant mode?
√
c) If x(0) = (π, 17)T , ﬁnd the limit, as t → ∞, of x1 (t)/x2 (t).
3. Consider the NONLINEAR system of diﬀerential equations
dx1
= 2(x2 − 1)
2
dt
dx2
= (x2 − 1)/2
1
dt
a) Find the ﬁxed points (there are four of them).
b) For each ﬁxed point, approximate the deviations from that ﬁxed point by
a linear system of equations.
c) Which ﬁxed points are stable? Unstable? Borderline?
4. In R3 , with the usual inner product, consider the vectors b1 = (1, 2, 2)T ,
b2 = (4, 4, 3)T , b3 = (2, 9, −1)T .
a) Use the GramSchmidt process to convert this basis for R3 into an orthogonal basis for R3 .
b) If v = (5, 2, 9)T , compute Pb1 v.
5. On R3 , let b1 = (3, 4, 5)T , b2 = (5, −10, 5)T , b3 = (7, 1, −5)T . Note that
these vectors are orthogonal. Let v = (2, 1, 2)T .
a) Write v as a linear combination of b1 , b2 , and b3 .
b) Let M be a matrix with eigenvalues λ1 = 5, λ2 = 3, λ3 = 15 and eigenvectors b1 , b2 , and b3 . Compute M v.
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 Spring '08
 RAdin
 Linear Algebra, Algebra, Eigenvectors, Vectors, differential equation dx, neutrally stable modes

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