Introduction_to_Linear_Systems

# Introduction_to_Linear_Systems - -1 1 2 1 4 1 4 2 1 1 2 1...

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Introduction to Linear Systems 1. Which of the following equations are linear? 1.1) 5 4 3 2 = + - z y x 1.2) 3 2 = + + z xy x 1.3) 4 3 2 = - + z y x 1.4) 1 2 = - z xy 1.5) 0 5 = - + + z y x 1.4) 3 2 2 = - + z y x π Answer: 1.1 and 1.4 2. Find the parametric solution to each equation, and then give two particular solutions. 2.1) 1 2 = + - z y x 2.2) 0 5 2 3 = - + - z y x 2.3) 2 2 3 = - y x Answer: 2.1) ) 2 , 1 , 0 ( ), 1 , 0 , 0 ( ), 2 1 , , ( t s z t y s x + - = = = 2.2) ) 1 , 0 , 3 ( ), 0 , 0 , 5 ( ), , , 2 3 5 ( t z s y t s x = = - + = 2.3) ) 2 , 2 ( ), 1 , 0 ( ), 1 2 3 , ( - - = = s y s x 3. Show that = = + + = - - = t x s x s t x s t x 4 3 2 1 1 1 is a solution to the system - = - + - - = + + - 3 3 2 3 3 2 4 3 2 1 4 3 2 1 x x x x x x x x 4. Determine whether the given set is a solution for the corresponding system 4.1) ( ) 2 , 0 , 3 for - = + - - = - + = - + 1 2 4 4 2 1 2 z y x z y x z y x 4.2) ( ) 1 , 1 , 2 - for = + + = - + = + + 2 2 2 3 6 9 3 z y x z y x z y x 4.3) ( ) 1 , 2 - for = + - = - 7 4 2 10 4 3 y x y x 4.4) ( ) 1 , 1 - for = - = - 4 3 1 3 y x y x Answer: 4.1) Yes, 4.2) Yes, 4.3) No, 4.4) No 5. Which of the systems in the previous problem are consistent? Answer: 4.1) and 4.2) 6. Find the augmented matrix of each system in problem 4. Answer: 4.1)

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Unformatted text preview: -----1 1 2 1 4 1 4 2 1 1 2 1 , 4.2) -2 1 1 1 2 1 1 2 3 6 9 3 , 4.3) --7 4 2 10 4 3 , 4.4) --4 3 1 1 3 1 7. Write a system of linear equations corresponding to each augmented matrices. 7.1) ---4 3 2 1 1 2 1 7.2) ----2 5 2 4 1 1 2 1 5 4 1 3 7.3) --2 1 1 1 4 1 3 2 Answer: 7.1) =-=--4 3 2 1 2 z y z y x 7.2) = +--= + +-= + +-2 5 2 4 1 2 5 4 3 w z x w y x w z y x 7.3) = = +-+ = + +-2 4 3 2 w w z y w z y x...
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Introduction_to_Linear_Systems - -1 1 2 1 4 1 4 2 1 1 2 1...

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