MATH 51 MIDTERM 1January 29, 20041. Find all solutions of the following system:x1− x2+ x3+ 2x4= 3x2+ x3+ x4= 3x1+ x2+ 3x3+ 4x4= 92. Let L be the intersection of the two planesx + y + z = 4 and 2x + 3y + z = 9.Find a parametric equation for L.3(a) Suppose u, v, and w are points in Rnsuch that kuk = kvk = kwk = 1 andsuch that w = −u. Suppose also that v is not equal to u or to w. Prove that thetriangle ∆uvw has a right angle at v.3(b) Suppose x, y, and z are vectors in Rnwhose norms are 1, 2, and 3, respec-tively. Suppose each vector is orthogonal (i.e., perpendicular) to each of the othertwo. Find a scalar c such that the vectorx + cy − zis orthogonal to the vector x + y + z.4. Consider the points A = (1, 1, 1, 1), B = (1, 2, 0, −1) and C = (1, 0, −1, 1) inR4.4(a) Find the cosine of the angle at B of the triangle ABC.4(b) Find a parametric equation for the plane through the points A, B, and Cfrom part (a).5. Are the following three vectors in R3linearly independent or linearly dependent?Show your work and explain your answer.u =123v =−21−1w =−1876. LetA =1 2 12 4 21 3 31 1 −1.1What 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=?.)7(a) Suppose v1, v2, . . . , vkare linearly dependent vectors in Rn. Show that ifA is an m × n matrix, then the vectors Av1, Av2, . . . , Avkmust also be linearlydependent.7(b) Suppose x, y , and z are linearly independent vectors in Rn. Prove that x,x + y, and x + y + z are also linearly independent.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) Find a basis for the column space C(A) of A.8(b) Find a basis for the nullspace N (A) of A.8(c) If v =12121, then Av =71028. (You do not need to check this.) Find allsolutions x ofAx =71028.9. Letx =112y =22−3A =·0 2 −21 1 4¸B =2 31 1−1 2Compute each the following:(a) 3x − 5y(b) x · (y + x)(c) kx − yk22(d) Ay(e) B(Ax)10(a,b,c). Suppose V is a set of vectors in Rn. What three properties must Vhave in order to be a linear subspace of V ?10(d,e). State whether each of the following sets is a linear subspace of R2. If itis not, explain why not.(d). The set W of vectors·xy¸such that x ≥ 0.(e). The set U of vectors·xy¸such that x is an
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