Math 51 - Winter 2011 - Midterm Exam IName:Student ID:Circle your section:Nick Haber James Zhao Henry Adams11:00 AM 10:00 AM 11:00 AM1:15 PM 1:15 PM 1:15 PMRalph Furmaniak Jeremy Miller Ha Pham11:00 AM 11:00 AM 11:00 AM1:15 PM 2:15 PM 1:15 PMSukhada Fadnavis Max Murphy Jesse Gell-Redman10:00 AM 11:00 AM 1:15 PM1:15 PM 1:15 PMSignature:Instructions: Print your name and student ID number, select thetime at which your section meets, and write your signature to in-dicate that you accept the Honor Code. There are 10 problemson the pages numbered from 1 to 12. Each problem is worth 10 points.In problems with multiple parts, the parts are worth an equal numberof points unless otherwise noted. Please check that the version of theexam you have is complete, and correctly stapled. In order to receivefull credit, please show all of your work and justify your answers. Youmay use any result from class, but if you cite a theorem be sure toverify the hypotheses are satisfied. You have 2 hours. This is aclosed-book, closed-notes exam. No calculators or other electronic aidswill be permitted. GOOD LUCK!Question 1 2 3 4 5 6 7 8 9 10 TotalScore1. Complete the following definitions.(a). A set {v1, v2, . . . , vk} of vectors in Rnis called linearly indepen-dent providedSolution(b). A function T : Rn→ Rkis called a linear transformation provided(c). A set S = {v1, . . . , vk} of vectors in a subspace V is called a basisfor V provided(d). A set V of vectors in Rnis called a subspace of Rnprovided(e). The dimension of a subspace V is12. Find the row reduced echelon form rref(A) of the matrixA =0 1 0 1 20 2 0 2 62 4 100 10 8.23. Consider the following matrix A and its row reduced echelon formrref(A):A =1 0 1 −1 00 1 2 3 00 1 2 0 −31 0 1 2 3, rref(A) =1 0 1 0 10 1 2 0 −30 0 0 1 10 0 0 0 0(You do not need to check that the row reduction is correct).(a). Find a basis for the column space C(A).(b). Find a basis for the nullspace N(A).34. Consider the matrix M =1 3 52 4 63 5 z. For which values of z will thesystem Mx =91011have:(a). (2 points) A unique solution? (Show your work below.)(b). (2 points) An infinite number of solutions?(c). (2 points) No solutions?Show your work here:44(d). (4 points) For z = 7, find the complete solution to the systemMx =91011.55. Let V be the set of all vectors x in R5that are orthogonal tou =11111and to v =−1−2−3−4−5. (To be in V , a vector must be orthogonalboth to u and to v.) Find a basis for V .66(a). Suppose that A is an m × n matrix of rank n. Find all thesolutions v of Av = 0. Explain your answer.6(b). Suppose that A is an m × n matrix of rank n as in part (a).Suppose v1, v2and v3are vectors such that Av1, Av2and Av3arelinearly dependent. Prove that the vectors v1, v2, and v3must also belinearly dependent.77(a). Find a parametric equation for the line L passing through thepoints A = (0, 4, 1) and B = (1, 3, 1).7(b). Find a point C on L such that the triangle ∆OAC has a rightangle at C. (Here O = (0, 0, 0) is the origin.)88(a). Let T : R2→ R2be a linear transformation with T12=−15and T11=23. Find a matrix A for T such that T (x) = Axfor all x ∈ R2. [Hint: What is12−11?]98(b). Let ∆ABC be a 3-4-5 right triangle in R2as shown below. LetS : R2→ R2be the rotation about the origin such thatS−→AB=35−→AC.FInd the matrix M such that S(x) = Mx for all x ∈ R2.3B4C5A109(a). Consider the points A = (2, 1, 3, 1), B = (4, 1, 5, 1) and C =(2, 3, 5, 1) in R4. Find a parametric equation for the plane through thepoints A, B, and C.9(b). Consider the triangle ABC (where A, B and C are the pointsgiven in part (a)). Find the cosine of the angle between the two sidesAB and AC.1110(a). (3 points) Consider the set V = {(x1, x2) ∈ R2|x1≤ 0, x2≤ 0}.Is V a linear subspace of R2? Explain.(b). (3 points) Suppose that T : R2→ R2is a linear transformationwith matrix B =2 00 −2and that x is a unit vector in R2. What, ifanything, can you conclude about the length of the vector T (x)?(c). (4 points) Suppose that u, v, and w are three linearly independentvectors. Show that u + v and u − v are linearly
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