MATH 2211 Midterm: Exam3210Sample3
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The properties of determinants: an n n matrix a is called orthogonal if aat = i. If a is orthogonal show that det(a) = 1: an n n matrix a is called skew-symmetric if at = a. Show that if a is skew-symmetric and n is an odd positive integer, then a is not invertible. Find the volume of the parallelepiped with one vertex at the origin and adjacent vertices are (1, 0, 2), (1, 2, 4), and (7, 1, 0). De ne the linear transformations t : p2 r2 by. Show that t is a linear transformation. b. ) Find a polynomial p in p2 that spans the kernel of t, and describe the range of t. Determine whether the set of polynomials is linearly independent or linearly dependent: p1(x) = 1, p2(x) = 2 + 4x2, p3(x) = 2x, and p4(x) = 12x + 8x3.
