Math Fall 2012 Exam 1 UMKC. Name. Student ID. Instructions: (a) The use of laptop or computer is prohibited.

Transcription

1 Math - Fall Exam UMKC Name Student ID Instructions: (a) The use of laptop or computer is prohibited. (b) Total time allowed for the exam: 75 min. (c) Calculators may not be shared. (d) For Part (Problems -), enter your choices at end of this page. (e) For Part (Problems -4), show all steps (calculators can be used only to check the final answers); unjustified answers will not receive credit; and clearly indicate your final answer. () () () (4) (5) () (7) (8) (9) ()

2 - 5 ) The augmented matrix for a system of equations is given by consistent, find the general solution. Otherwise state that there is no solution.. If the system is ) A) No solution B) x = -x - x + x4 - x5 + 5 x is free x is free x4 = 4 x 5 - x5 = -4 C) x = -x - x + 9 x is free x = -4 x4 = 4 x 5 - x5 = -4 D) x = -x - x + 9 x is free x is free x4 = -4 x5 = -4 ) Let A = and b = b b b. ) Determine if the equation Ax = b is consistent for all possible b, b, b. If the equation is not consistent for all possible b, b, b, give a description of the set of all b for which the equation is consistent (i.e., a condition which must be satisfied by b, b, b ). A) Equation is consistent for all b, b, b satisfying -b + b + b =. B) Equation is consistent for all b, b, b satisfying b + b + b =. C) Equation is consistent for all b, b, b satisfying -b + b =. D) Equation is consistent for all possible b, b, b. ) For what values of h are the given vectors linearly dependent? - 4, 5 -,, -4-8 h A) Vectors are linearly dependent for all h B) Vectors are linearly independent for all h C) Vectors are linearly dependent for h -4 D) Vectors are linearly dependent for h = -4 ) 4) Let T: -> be a linear transformation that maps u = - 4 into - and maps v = 4 into 4) -8. Use the fact that T is linear to find the image of u + v. A) B) C) D) -

3 5) Let u = - 5 and v = -. Display the vectors u, v, and u + v on the same axes. 5) A) B) C) D)

4 ) Let A = and b = Define a transformation T: -> by T(x) = Ax. If possible, find a vector x whose image under T is b. Otherwise, state that b is not in the range of the transformation T. A) -5-5 B) - C) D) b is not in the range of the transformation T. ) 7) The columns of I = are e =, e =, e =. 7) Suppose that T is a linear transformation from into such that T( e) = -, T( e ) =, and T( e ) = -. x Find a formula for the image of an arbitrary x = x in. x A) C) T T x x x x x x = x + x- x -x + x = x- x x x + x B) T D) T x x x x x x = x - x x = x + x - x x -x + x 8) Let T be the linear transformation whose standard matrix is - A = Determine whether the linear transformation T is one-to-one and whether it maps onto. A) One-to-one; not onto B) One-to-one; onto C) Not one-to-one; not onto D) Not one-to-one; onto 8)

5 9) Let A= and B= A) Yes B) No. Determine whether A and B are inverses of each other. 9) ) Let A = 7 8. Find the inverse of the matrix A, if it exists. ) A) A- = B) A- = 7 8 C) A- = D) A- does not exist.

6 PART II WORK OUT Directions: Present your solutions in the space provided. Show all your work neatly and concisely and Box your final answer. You will be graded not merely on the final answer, but also on the quality and correctness of the work leading up to it. ) [ points] Let A=, B= and C =. Compute the following expressions. a) (A - B T ) T + C b) (A - C - ) - + (C T +B T ) T

7 ) [ points] Solve the system of linear equations using the Gauss-Jordan elimination method. x + y - z = x - y - z = - x + 4y -z = 4

8 ) [ points] (a) Write a system of equations that determines the loop currents. (b) Form the augmented matrix related to the system. Note: There is no need to solve the system.

9 4) [ points] Let T(x, x, x ) = (x -5 x + 4 x, x - x ). (a) Show that T is a linear transformation by finding a matrix that implements the mapping. (b) Does T map R onto R? Justify your answer. (c) Is T a one-to-one mapping? Justify your answer.