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Midterm Exam 2 Review Problems - First-Year Interest Group Seminar | N 1, Exams of Health sciences

Material Type: Exam; Class: FIRST-YEAR INTEREST GROUP SMNR; Subject: Nursing; University: University of Texas - Austin; Term: Fall 2005;

Typology: Exams

Pre 2010

Uploaded on 08/30/2009

koofers-user-9s4-2
koofers-user-9s4-2 🇺🇸

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M346 Second Midterm Exam, November 10, 2005

  1. Let A =

( 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.

  1. a) In R^3 with the standard inner product, apply the Gram-Schmidt process

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).

  1. Consider the system of difference equations x(n + 1) = Ax(n), where

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.)

  1. Consider the nonlinear system of differential equations:

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.

  1. a) Find a least-squares solution to Ax = b, where A =

  

  

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^.