MATH 51 MIDTERM (FEBRUARY 2, 2010)Max Murphy Jonathan Campbell Jon Lee Eric Malm11am 11am 10am 11am1:15pm 2:15pm 1:15pm 1:15pmXin Zhou Ken Chan (ACE) Jose Perea Frederick Fong11am 1:15pm 11am 11am1:15pm 1:15pm 1:15pmYour name (print):Sign to indicate that you accept the honor code:Instructions: Find your TA’s name in the table above, and circlethe time that your TTh section meets. During the test, you may notuse notes, books, or calculators. Read each question carefully, andshow all your work. Each of the 10 problems is worth 10 points. Youhave 90 minutes to do all the problems.1.2.3.4.5.6.7.8.9.10.Total11. Complete the following definitions.(a). A set {v1, v2, . . . , vk} of vectors in Rnis called linearly dependentprovided(b). A set V of vectors in Rnis called a linear subspace provided(c). A map T : Rn→ Rkis called a linear map provided(d). A set S = {v1, . . . , vk} of vectors in a linear subspace V is calleda basis for V provided(e). The dimension of a subspace V is22. Find the row reduced echelon form rref(A) of the matrixA =0 1 0 1 20 2 0 2 52 4 7 10 8.33(a). Consider the following matrix B and its row reduced echelonform rref(B):B =4 3 7 0 32 3 5 0 21 1 2 0 15 4 9 0 4, rref(B) =1 0 1 0 00 1 1 0 00 0 0 0 10 0 0 0 0(You do not need to check this.) Find a basis for the column spaceC(B) of B.3(b). Find a basis for the nullspace N (B) of B (where B is as in part(a)).44(a). Let A =1 32 71 23 10. Find the condition(s) on a vector b for bto be in the column space of A. (Your answer should be one or moreequations involving the components biof b.)54(b). Find a matrix B such that N(B) = C(A). (Here A is the matrixin part (a).)65. Let V be the set of all vectors x in R4that are orthogonal tou =1111and to v =2234. (To be in V , a vector must be orthogonalboth to u and to v.) Find a basis for V .76(a). Suppose u and v are vectors in Rnsuch that kuk = kvk. Provethat the vectors u − v and u + v are orthogonal to each other.6(b). Suppose that v1, v2, and v3are linearly dependent vectors inRn. Suppose that A is an m × n matrix. Prove that the vectors Av1,Av2, and Av3must also be linearly dependent.87(a). Find a parametric equation for the line L through the pointsA = (0, 1, 1) and B = (1, 2, 3).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.)98. Suppose T : R3→ R2is a linear transformation such thatT100=31, T110=713, T112=720.Find the matrix for T .109. Consider the points A = (1, 1, 1, 1), B = (1, 2, 0, 1) and C =(1, 0, 1, 1) in R4.9(a). Find the cosine of the angle at B of the triangle ABC.9(b). Find a parametric equation for the plane through the points A,B, and C.1110. Short answer questions. (No explanations required.)(a). Suppose that a linear subspace V is spanned by vectors v1, v2, . . . , vk.What, if anything, can you conclude about the dimension of V ?(b). Suppose that a linear subspace W contains a set {w1, w2, . . . , wk}of k linearly independent vectors. What, if anything, can you concludeabout the dimension of W ?(c). Suppose u · v < 0. What, if anything, can you conclude about theangle θ between u and v ? [Note: by definition, the angle θ betweentwo nonzero vectors is in the interval 0 ≤ θ ≤ π.](d). Suppose T : Rk→ Rnis a linear map and b ∈ Rn. If k < n,what, if anything, c an you conclude about the number of solutions ofAx = b?(e). Suppose V is a 3 dimensional linear subspace of R6and supposethat v1, v2, and v3are linearly independent vectors in V . What more,if anything, must you know in order to conclude that {v1, v2, v3} is abasis for V
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