129a_proj1_sol - MATH 129A Linear Algebra I Project 1...

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Unformatted text preview: MATH 129A Linear Algebra I Project 1 solution 1. (a) x1 = the size of the field producing x2 = the size of the field producing (b) x1 + x2 = 5349 3 2 x1 + x2 = 3733 4 3 (c) x1 = 2004, 2. (a) There are 3 unknowns and they are x1 = number of measures in a bundle of first type of corn x2 = number of measures in a bundle of second type of corn x3 = number of measures in a bundle of third type of corn (b) There are 3 equations and they are x2 = 3345 3 4 2 3 bushel per square yard bushel per square yard 3x1 + 2x2 + x3 = 368 x1 + 3x2 + 2x3 = 349 2x1 + x2 + 3x3 = 447 (c) The augmented matrix is 3 2 1 368 1 3 2 349 2 1 3 447 (d) The RREF is 1 0 0 71 0 1 0 32 0 0 1 91 (e) x1 = 71, x2 = 32, x3 = 91 3. (a) The linear system is f1 + f2 = 24 f1 + f3 = 12 f3 + f4 = 24 f2 + f4 = 36 (b) The general solution in parametric form is f1 = f4 - 12 f2 = 36 - f4 f3 = 24 - f4 f4 = free (c) maximum flow is 24, and minimum flow is 12 (d) When f4 = 18, from (b), we have f1 = 6, f2 = 18, f3 = 6 4. (a) Label the temp of the interior nodes as follows: x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13 x14 x15 x16 Then the augmented matrix of the linear system is 4 -1 0 0 -1 0 0 0 0 0 0 0 0 0 0 0 50 -1 4 -1 0 0 -1 0 0 0 0 0 0 0 0 0 0 35 0 -1 4 -1 0 0 -1 0 0 0 0 0 0 0 0 0 35 0 0 -1 4 0 0 0 -1 0 0 0 0 0 0 0 0 55 -1 0 0 0 4 -1 0 0 -1 0 0 0 0 0 0 0 15 0 -1 0 0 -1 4 -1 0 0 -1 0 0 0 0 0 0 0 0 0 -1 0 0 -1 4 -1 0 0 -1 0 0 0 0 0 0 0 0 0 -1 0 0 -1 4 0 0 0 -1 0 0 0 0 20 0 0 0 0 -1 0 0 0 4 -1 0 0 -1 0 0 0 15 0 0 0 0 0 -1 0 0 -1 4 -1 0 0 -1 0 0 0 0 0 0 0 0 0 -1 0 0 -1 4 -1 0 0 -1 0 0 0 0 0 0 0 0 0 -1 0 0 -1 4 0 0 0 -1 20 0 0 0 0 0 0 0 0 -1 0 0 0 4 -1 0 0 40 0 0 0 0 0 0 0 0 0 -1 0 0 -1 4 -1 0 25 0 0 0 0 0 0 0 0 0 0 -1 0 0 -1 4 -1 25 0 0 0 0 0 0 0 0 0 0 0 -1 0 0 -1 4 45 (b) (i) highest temp is 28.7308F, lowest temp is 19.7917F (ii) the average temp is 23.75F ...
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