MATH 51 MIDTERM 1October 16, 2008Professor: Han Kargin White Wise TTh Section Number:TA:Olena Bormashenko Luis Diogo Kaveh Fouladgar Frederick Tsz Ho FongRobin Koytcheff Jason Lo Jonathan Lee Jose PereaJosh GenauerTime your TTh section meets: morning afternoonYour name (print): Student ID:Sign to indicate that you accept the honor code:Instructions: Circle your professor’s name, your TA’s name, and the time thatyou attend the TTh section. During the test, you may not use notes, books, orcalculators. Read each question carefully, and show all your work. Each of the nineproblems is worth 10 points. You have 90 minutes to do all the problems.123456789Total1Name:1. Find all solutions of the following system:x1+ 2x2+ x3+ x4= 7x1+ 2x2+ 2x3− x4= 122x1+ 4x26x4= 4.2Name:2(a). Find a parametric equation for the plane containing the points A = (1, 2, 3),B = (4, 5, 6), and C = (2, 2, 3).2(b). Find the equation for the plane that passes through the point A = (1, 2, 3)and that is perpendicular to the vector v =735. (Your answer should be anequation of the form ax + by + cz = d.)3Name:3(a) Suppose that ∆ is an equilateral triangle in R3and that the edges of ∆ eachhave length 1. Let A, B, and C be the vertices of ∆. Find(3−→AB) · (5−→AC).4Name:3(b). Suppose A =a1a2a3a4and B =b1b2b3b4are orthogonal vectors in R4witha4> 0 and b4> 0.Let a =a1a2a3and b =b1b2b3.Prove that the angle between a and b is obtuse (i.e., greater than π/2).5Name:4. Let V be the set of vectors in R4that are orthogonal to the vector a =1205.Find a basis for V .6Name:5. Are the following three vectors in R3linearly independent or linearly dependent?Show your work and explain your answer.u =212v =429w =−2−23.7Name:6. LetA =1 2 12 4 21 3 31 1 −1.6(a). What condition(s) must b satisfy to be in the column space of A?(Your answer should be one or more equations of the form ? b1+ ? b2+ ? b3+ ? b4=?.)8Name:6(b) Find a matrix M such that the column space of A is equal to the null spaceof M. [Hint: use your answer to part (a).]9Name:7(a) Suppose x, y, and z are linearly independent vectors in Rn. Prove that thevectors x + y, x − y, and x + y + z are also linearly independent.10Name:7(b) Suppose that v1, v2, v3, v4are linearly dependent vectors in Rn, and thatv1, v2, v3are linearly independent.Prove that v4is a linear combination of v1, v2, and v3.11Name:8. Let A be the matrixA =1 1 1 0 31 1 2 1 30 0 0 2 −21 1 1 1 2The reduced echelon form for A isR =1 1 0 0 20 0 1 0 10 0 0 1 −10 0 0 0 0(You do not need to check this.)8(a) (3 points) Find a basis for the column space C(A) of A.12Name:8(b) (4 points) Find a basis for the nullspace N(A) of A.13Name:8(c) (3 points) Find all solutions x ofAx =1201+0121.[Hint: compare the right hand side of this equation to the columns of A.]14Name:9(a,b,c). Suppose V is a set of vectors in Rn. What three properties must V havein order to be a linear subspace of V ?9(d,e). Suppose that V is a linear subspace of Rnand that v1, v2, . . . , vkarevectors in Rn. What two properties must v1, v2, . . . , vkhave in order to be a basisfor V
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