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Material Type: Exam; Class: FIRST-YEAR INTEREST GROUP SMNR; Subject: Nursing; University: University of Texas - Austin; Term: Fall 2005;
Typology: Exams
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M346 Second Midterm Exam, November 10, 2005
( 2 − 1 2 − 1
) .
a) Find the eigenvalues and eigenvectors of A.
b) Compute A^13421. (No, you do NOT need a calculator for this!)
c) Compute eA.
to convert the basis
{ (1, 4 , 3)T^ , (2, 3 , 4)T^ , (10, 4 , 0)T^
} into an orthogonal basis.
b) Find the coordinates of the vector v = (1, − 7 , 9)T^ in the orthogonal basis you constructed in part (a).
A =
and^ x(0) =
.
a) Diagonalize A.
b) Find x(n) for all n. (You may express your answer as a linear combination of the eigenvectors of A, but the coefficients should be explicit.)
dx 1 dt
= ln(x 1 x^22 )
dx 2 dt
= ln(x^21 x 2 ).
a) Find the fixed point (there is only one).
b) Find a LINEAR system of ODEs that approximates motion near the fixed point.
c) Find all the stable modes of this linear system. Then find all the unstable modes.
and b =
b) Find the equation of the best line through the points (− 1 , 3)T^ , (0, 1)T^ , (1, 0)T^ , and (2, −2)T^.