线性代数及其应用习题解答(3e)_Gilbert Strang.pdf

INTRODUCTION TO LINEAR ALGEBRA Third Edition MANUAL FOR INSTRUCTORS Gilbert Strang gs@ Massachusetts Institute of Technology /18.06/www /˜gs Wellesley-Cambridge Press Box 812060 Wellesley, Massachusetts 02482 Solutions to Exercises Problem Set 1.1, page 6 1 Line through (1, 1, 1); plane; same plane! 3 v = (2, 2) and w = (1, −1). 4 3v + w = (7, 5) and v − 3w = (−1, −5) and cv + dw = (2c + d, c + 2d). 5 u + v = (−2, 3, 1) and u + v + w = (0, 0, 0) and 2u + 2v + w = (add first answers) = (−2, 3, 1). 6 The components of every cv + dw add to zero. Choose c = 4 and d = 10 to get (4, 2, −6). 8 The other diagonal is v − w (or else w − v ). Adding diagonals gives 2v (or 2w ). 9 The fourth corner can be (4, 4) or (4, 0) or (−2, 2). 10 i + j is the diagonal of the base. 11 Five more corners (0, 0, 1), (1, 1, 0), (1, 0, 1), (0, 1, 1), (1, 1, 1). The center point is ( 1 , 1 , 1 ). The 2 2 2 centers of the six faces are ( 1 , 1 , 0), ( 1 , 1 , 1) and (0, 1 , 1 ), (1, 1 , 1 ) and ( 1 , 0, 1 ), ( 1 , 1, 1 ). 2 2 2 2 2 2 2 2 2 2 2 2 12 A four-dimensional cube has 24 = 16 corners and 2 · 4 = 8 three-dimensional sides and 24 two-dimensional faces and 32 one-dimensional edges. See Worked Exam