MA 242 ExamsFall 2008Here are the four exams that I gave in my MA 242 class in the Fall of 2008. You shoulduse them to get an idea of the format of a typical test and to see the types of questions Iask. You should not assume that the test questions this semester will be on the same topics.In fact, you are always responsible for all of the material that we cover in class as well asall of the designated material from your text. The best way to study for my exams is to besure that you are very comfortable with the homework assignments and the examples thatI present in class. My tests often vary in difficulty (as you can see here), and your gradefor the examination will be determined by a curve that will be announced in class after theexamination is graded.MA 242 Exam 1A October 1, 2008Name:Directions: Please do all of your work in this exam booklet and make sure that you crossout any work that we should ignore when we grade. Books and extra papers are notpermitted. If you have a question about a problem, please ask. Remember: answers thatare written logically and clearly will receive higher scores. There are 5 questions on 6 pages(not counting this cover page). Please make sure that you have all 6 pages of questions.Do not write in the following box:PROBLEM POSSIBLE SCORE1 152 163 214 205 28TOTAL 100MA 242 Exam 1A October 1, 20081. (15 points) Row reduce the matrixA =1 2 1 2−1 −2 0 −22 4 −1 6to reduced row echelon form (RREF). Do only one row operation at a time andspecify that operation when you perform it. Indicate when you first arrive at a matrixin echelon form (REF). What are the pivot positions of A?1MA 242 Exam 1A October 1, 20082. (16 points) Find the value(s) of h such that the following set of vectors is linearlyindependent:1−1−3,−578,11h2MA 242 Exam 1A October 1, 20083. (21 points) Which of the following functions T : R2→ R2are linear? If the function islinear, what is its standard matrix representation? If the function is not linear, justifyyour answer by giving an example of one of the linearity properties that does not hold.(a) T (x1, x2) = (3x1− x2, 2|x1|)(b) T (x1, x2) = (x2− x1, sin x2)(c) T (x1, x2) = (3x1− x2, 2x1+ 4x2)3MA 242 Exam 1A October 1, 20084. (20 points) Given the matrixA =1 2 −7 50 1 −4 01 0 1 02 −1 6 8let T : R4→ R4be the linear transformation defined by T (x) = Ax. Find all vectorsx in R4such thatT (x) =8119.4MA 242 Exam 1A October 1, 20085. (28 points) Are the following statements true or false? You will not receive anycredit unless you justify your answers. (Note that there are two more parts tothis question on the next page.)(a) Each matrix is row equivalent to a unique matrix in echelon form.(b) An m × n matrix A defines a linear transformation T : Rm→ Rnby the formulaT (x) = Ax.5MA 242 Exam 1A October 1, 2008Question 5 (continued):(c) The columns of any 4 × 5 matrix are linearly dependent.(d) Suppose that A is an m × n matrix with m ≥ 2 and B is an n × p matrix. Thenthe second row of AB is the second row of A multiplied on the right by B.6MA 242 Exam 2A October 29, 2008Name:Directions: Please do all of your work in this exam booklet and make sure that you crossout any work that we should ignore when we grade. Books and extra papers are notpermitted. If you have a question about a problem, please ask. Remember: answers thatare written logically and clearly will receive higher scores. There are 5 questions on 7 pages(not counting this cover page). Please make sure that you have all 7 pages of questions.Do not write in the following box:PROBLEM POSSIBLE SCORE1 182 183 164 205 28TOTAL 100MA 242 Exam 2A October 29, 20081. (18 points) Compute the determinant of the matrixA =2 3 0 1−2 0 1 −11 0 2 20 −4 −1 0using cofactor expansion until you arrive at 2×2 matrices. You may check your answerusing your calculator, but you will not receive any credit unless you show all steps inthe computation.1MA 242 Exam 2A October 29, 20082. (18 points) Use Cramer’s Rule to compute the solution to the system2x1+ x2= 7−3x1+ x3= −8x2+ 2x3= −3.2MA 242 Exam 2A October 29, 20083. (16 points) Find the matrix A whose inverse is A−1="4 56 7#.3MA 242 Exam 2A October 29, 20084. (20 points) Recall that Pnis the vector space of all polynomials p(t) of degree at most n.Let L : P2→ P3be the transformation given byL(p(t)) = (t + 2) p(t).(a) Show that L is a linear transformation.4MA 242 Exam 2A October 29, 2008Question 4 (continued):(b) What are the kernel and the range of L? Describe the range of L as a singleequation for the coefficients a3, a2, a1, and a0of p(t) = a3t3+ a2t2+ a1t +a0. Thisequation should not include the variable t. Provide a brief justification for bothof your answers.5MA 242 Exam 2A October 29, 20085. (28 points) Are the following statements true or false? You will not receive anycredit unless you justify your answers. (Note that there are two more parts tothis question on the next page.)(a) Every elementary matrix is invertible.(b) If A and B are row equivalent square matrices, then det A = det B.6MA 242 Exam 2A October 29, 2008Question 5 (continued):(c) Row operations on a matrix can change the null space.(d) If A is a square matrix and the equation Ax = e1has a unique solution, then Ais invertible.7MA 242 Exam 3A December 10, 2008Name:Directions: Please do all of your work in this exam booklet and make sure that you crossout any work that we should ignore when we grade. Books and extra papers are notpermitted. If you have a question about a problem, please ask. Remember: answers thatare written logically and clearly will receive higher scores. There are 5 questions on 6 pages(not counting this cover page). Please make sure that you have all 6 pages of questions.Do not write in the following box:PROBLEM POSSIBLE SCORE1 182 183 184 185 28TOTAL 100MA 242 Exam 3A December 10, 20081. (18 points) Consider the subspace H of R4given bya + 2b − 2c + d2a + 4b − 3c + 6d−3a − 6b + 7c + dc + 4d: a, b, c, d in R.Find a basis for H. What is the dimension of H?1MA 242 Exam 3A December 10, 20082. (18 points) LetA =−4 −2 2 1−2 −1 −3 −43 −6 2 10 0 1 −7.Compute a basis for the λ = −5