Math294Prelim_1.sol.1.ver3.Spring_2008

# Math294Prelim_1.sol.1.ver3.Spring_2008 - Math 294 Problem 1...

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Math 294 Prelim 1 Solutions Spring 2008 Problem 1. 1a The row reduction process is: ( 29 ( 29 ( 29 1 0 2 4 1 1 0 2 4 1 1 0 2 4 1 0 1 3 1 0 0 1 3 1 0 0 1 3 1 0 3 4 6 8 0 4 12 4 3 0 1 3 1 3 / 3 0 1 3 4 0 0 1 3 4 0 0 1 3 4 0 1 1 0 2 4 1 1 0 2 4 0 0 1 3 1 0 0 1 3 1 0 0 0 0 0 3 / 3 0 0 0 1 3 / 3 0 0 0 3 0 0 0 0 0 1 0 b b b b b - - - - - - - - - - - - - - - - - - - - - - ( 29 1 2 0 0 0 1 3 0 0 0 0 0 1 3 / 3 0 0 0 0 b - - 1b. There are 3 leading ones in rref( A ), so the rank of A is 3. 1c The system is consistent if and only if 3 b = . 1d 4 3 2 1 0, , 3 , 1 2 x x s x s x s = = = = - , or in vector form, 1 2 0 3 0 1 0 0 x s -       = +       r There are infinitely many solutions, since s is a free variable. 1e. The Kernel of A is the line 2 3 1 0 s - . It is a line through the origin in 4 4 4 R Problem 2. 1 2 3 1 2 3 2 3 3 x ax bx y cx dx y ex y + + = + = =

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( 29 ( 29 3 3 3 3 2 3 2 2 2 3 1 2 3 1 1 1 2 3 1 2 3 3 / / / / / / ex y x y e cx dx y x y c dy ce x ax bx y x y ax bx y a y c dy ce
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Math294Prelim_1.sol.1.ver3.Spring_2008 - Math 294 Problem 1...

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