MATH 51 MIDTERMOctober 20, 2005Name:Numeric Student ID:Instructor’s Name:I agree to abide by the terms of the honor code:Signature:Instructions: Print your name, student ID number and instructor’s name in the spaceprovided. During the test you may not use notes, books or calculators. Read eachquestion carefully and show all your work; full credit cannot be obtained withoutsufficient justification for your answer unless explicitly stated otherwise. Underline yourfinal answer to each question. There are 8 questions. You have 90 minutes to do all theproblems.Question Score Maximum1 102 153 104 105 106 107 108 10Total 85Question 5 of 8, Page 2 of 3 Solutions1. Solve the following system of equations in the variables x, y, z, w:x − y − z + w = 5y − z + 2w = 82x − y − 3z + 4w = 18If a solution exists, express your answer in parametric form.2. Let A be the matrix1 1 0 1 41 2 1 1 60 1 1 1 32 2 0 1 7.(a) Find a basis for the nullspace of A.(b) Find a basis for the column space of A.(c) Using your work from the previous page, what is the set of all solutions tothe equationAx =1102, where x =x1x2x3x4x5?State your answer in parametric form.3. Let P1be the plane described by normal vector (1, 1, 1) and containing the point(0, 0, 1). Let P2be the plane described by the equation x + 2y + 3z = 0.(a) Write P1as an equation of the form ax + by + cz = d.(b) What is the set of points in the intersection of P1and P2? If there are pointsin the intersection, express them in parametric form.4. The coordinates of three points P , Q and R are (1, 1, 1), (2, 1, 0) and (3, 2, 3)respectively.(a) Show that the vectors~P Q and~P R are perpendicular.(b) Determine the area of the triangle P , Q and R.5. For each of the following sets, determine whether or not the set is a subspace. Forthis question only, you do not need to show your work; simply write SUBSPACEor NOT SUBSPACE.Question 8 of 8, Page 3 of 3 Solutions(a) The setx1x2...xnin Rn: x1+ x2+ · · · + xn= 1.(b) The setx1x2...xnin Rn: xi≥ 0 for 1 ≤ i ≤ n.(c) The nullspace N (A), whereA =3 0 11 6 −17.(d) The setxy: y = x2.(e) The setx1x2...xnin Rn: x1+ x2+ · · · + xn= 0.6. Let {u, v, w} be a linearly independent set of vectors. Show that the set{u, u + 2v, u + 2v + 3w}is a linearly independent set of vectors, as well.7. LetA =5 −2 02 1 02 0 1, and x =x1x2x3.Find all solutions to the equation Ax = 3x.8. Let V be the set of all points P in R4such that the distance from P to each oneof the points (1, 2, 3, 4), (2, 3, 4, 1), and (3, 4, 2, 1) are all equal. Show that V is alinear subspace of R4and compute its dimension.Note: The distance between two points x = (x1, x2, . . . , xn) and y = (y1, y2, . . . , yn)in Rnis by definition:dist(x, y) =p(x1− y1)2+ (x2− y2)2+ . . . + (xn−
View Full Document
Unlocking...