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MATH 311–503/505 Fall 2015 Sample problems for the final exam Any problem may be altered or replaced by a different one! Problem 1 Find the point of intersection of the planes x + 2y − z = 1, x − 3y = −5, and 2x + y + z = 0 in R3 . Problem 2 Consider a linear operator L : R3 → R3 given by L(v) = (v · v1 )v2 , where v1 = (1, 1, 1), v2 = (1, 2, 2). (i) Find the matrix of the operator L. (ii) Find the dimensions of the range and the kernel of L. (iii) Find bases for the range and the kernel of L. Problem 3 Let v1 = (1, 1, 1), v2 = (1, 1, 0), and v3 = (1, 0, 1). Let L : R3 → R3 be a linear operator on R3 such that L(v1 ) = v2 , L(v2 ) = v3 , L(v3 ) = v1 . (i) Show that the vectors v1 , v2 , v3 form a basis for R3 . (ii) Find the matrix of the operator L relative to the basis v1 , v2 , v3 . (iii) Find the matrix of the operator L relative to the standard basis. 1 1 1 Problem 4 Let B = 1 1 1 . 1 1 1 (i) Find all eigenvalues of the matrix B. (ii) Find a basis for R3 consisting of eigenvectors of B. (iii) Find an orthonormal basis for R3 consisting of eigenvectors of B. (iv) Find a diagonal matrix D and an invertible matrix U such that B = UDU −1 . Problem 5 Let V be a subspace of R4 spanned by vectors x1 = (1, 1, 0, 0), x2 = (2, 0, −1, 1), and x3 = (0, 1, 1, 0). (i) Find the distance from the point y = (0, 0, 0, 4) to the subspace V . (ii) Find the distance from the point y to the orthogonal complement V ⊥ . Problem 6 Consider a vector field F(x, y, z) = xyze1 + xye2 + x2 e3 . (i) Find curl(F). (ii) Find the integral of the vector field curl(F) along a hemisphere H = {(x, y, z) ∈ R3 : 2 x + y 2 + z 2 = 1, z ≥ 0}. Orient the hemisphere by the normal vector n = (0, 0, 1) at the point (0, 0, 1). Problem 7 Find the area of a pentagon with vertices (0, 0), (4, 0), (5, 2), (3, 4), and (−1, 2).