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Name: CSU ID: Version 1, Exam 1, Math 369 September 25, 2013 Show all work to receive full credit. Part A 1. Given the matrix A and vector ~b below, find the following: (a) RREF of [A|~b] (the intermediate elimination steps need not be defined); (b) the number of free variables and what they are; (c) homogeneous solution of A~xH = ~0 with ~xH written in vector form; and (d) the general solution of A~x = ~b written in vector form with the particular solution and the homogeneous solution(s) in separate vectors. A= 2 −3 0 1 0 −4 6 3 0 1 8 −12 −9 −2 −2 2 −3 18 13 1 , ~b = 14 −4 −12 138 2. (a) Given A below, if the (1,1) position is to remain unchanged, define the operator, E1 , that leaves row 1 unchanged and eliminates column 1 from the second row to the third. 2 −3 4 9 −13 A = −4 8 −21 29 (b) Given that the LU decomposition of A is 1 0 0 2 −3 4 1 0 0 3 −5 A = LU = −2 4 −3 1 0 0 −2 and 37 ~b = −113 253 solve A~x = ~b using forward and backward substitution. The answer should be defined with fractions, if necessary. 3. Find a vector w ~ consisting of integers, such that it is perpindicular to the following vectors ~v1 = −1 2 −1 0 , ~v2 = −3 9 −3 2 , ~v3 = −1 −10 2 −8 Name: SI: Version 1, Exam 1, Math 369 September 25, 2013 Show all work to receive full credit. Part B 4. Refer to the matrix A in problem 2. (a) Find the cofactor C2,3 . (b) Using the LU decomposition, find det(A). (c) Using the information from (b), if B is the matrix produced by flipping columns 1 and 3 of A, what is det(B)? 5. Consider the vectors ~v1 and ~v2 in ]3. (a) Find the angle between ~v1 and ~v2 . Is it acute or obtuse? (b) Find proj ~v1 ~v2 (c) Find the length of proj~v1 ~v2 . (d) Find w ~ such that (1) ~v1 and w ~ are orthogonal and (2)~v2 = proj~v1 ~v2 + w ~