INTRODUCTION TO LINEAR ALGEBRA Third Edition MANUAL FOR INSTRUCTORS Gilbert Strang
[email protected] Massachusetts Institute of Technology http://web.mit.edu/18.06/www http://math.mit.edu/˜gs http://www.wellesleycambridge.com 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 ( 21 , 12 , 12 ). The centers of the six faces are ( 12 , 12 , 0), ( 12 , 21 , 1) and (0, 12 , 12 ), (1, 12 , 12 ) and ( 12 , 0, 12 ), ( 12 , 1, 12 ). 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 Example 2.4 A. 13 sum = zero vector; sum = −4:00 vector; 1:00 is 60◦ from horizontal = (cos π3 , sin π3 ) = ( 12 , √ 3 ). 2 14 Sum = 12j since j = (0, 1) is added to every vector. 15 The point 3 v 4 + 14 w is three-fourths of the way to v starting from w . The vector 1 v 4 + 14 w is halfway to u = 21 v + 12 w , and the vector v + w is 2u (the far corner of the parallelogram). 16 All combinations with c + d = 1 are on the line through v and w . The point