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Unformatted text preview: 4. Linear Systems 4A. Review of Matrices ⎝ ⎞ ⎠ 1 − 1 ⎞ ⎠ 2 1 1 4A1. Verify that 2 1 = . 1 − 1 − 2 3 − 2 − 6 − 1 2 ⎞ ⎠ ⎞ ⎠ 1 2 − 1 4A2. If A = and B = , show that AB = BA . 3 − 1 2 1 ⎞ ⎠ 4A3. Calculate A − 1 if A = 2 2 , and check your answer by showing that AA − 1 = I 3 2 and A − 1 A = I . 4A4. Verify the formula given in Notes LS.1 for the inverse of a 2 × 2 matrix. ⎞ ⎠ 1 4A5. Let A = . Find A 3 (= A · A · A ). 1 1 4A6. For what value of c will the vectors x 1 = (1 , 2 , c ) , x 2 = ( − 1 , , 1) , and x 3 = (2 , 3 , 0) be linearly dependent? For this value, find by trial and error (or otherwise) a linear relation connecting them, i.e., one of the form c 1 x 1 + c 2 x 2 + c 3 x 3 = 4B. General Systems; Elimination; Using Matrices 4B1. Write the following equations as equivalent firstorder systems: d 2 x dx y − x 2 a) + 5 + tx 2 = b) y + (1 − x 2 ) y = sin x dt 2 dt 4B2. Write the IVP y (3) + p ( t ) y + q ( t ) y + r ( t ) y = , y (0) = y , y (0) = y , y (0) = y as an equivalent IVP for a system of three firstorder linear ODE’s. Write this system both as three separate equations, and in matrix form. ⎞ ⎠ ⎞ ⎠ 1 1 x 4B3. Write out x = x , x = as a system of two firstorder equations. 4 1 y a) Eliminate y so as to obtain a single secondorder equation for x . b) Take the secondorder equation and write it as an equivalent firstorder system. This isn’t the system you started with, but show a change of variables converts one system into the other. 4B4. For the system x = 4 x − y, y = 2 x + y , 2 t a) using matrix notation, verify that x = e 3 t , y = e 3 t and x = e , y = 2 e 2 t are solutions; b) verify that they form a fundamental set of solutions — i.e., that they are linearly independent; 1 2 18.03 EXERCISES c) write the general solution to the system in terms of two arbitrary constants c 1 and c 2 ; write it both in vector form, and in the form x = . . . , y = . . . . ⎞ ⎠ 1 3 4B5. For the system x = A x , where A = 3 1 , ⎞ ⎠ ⎞ ⎠ 1 − 2 t a) show that x 1 = 1 e 4 t and x 2 = e form a fundamental set of solutions 1 − 1 (i.e., they are linearly independent and solutions); ⎞ ⎠ 5 b) solve the IVP: x = A x , x (0) = . 1 ⎞ ⎠ 1 1 4B6. Solve the system x = x in two ways: 1 a) Solve the second equation, substitute for y into the first equation, and solve it. b) Eliminate y by solving the first equation for y , then substitute into the second equation, getting a second order equation for x . Solve it, and then find y from the first equation. Do your two methods give the same answer?...
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This note was uploaded on 10/21/2011 for the course MATH 18.03 taught by Professor Vogan during the Fall '09 term at MIT.
 Fall '09
 vogan
 Linear Systems, Matrices

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