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**Unformatted text preview: **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 3 v + w = (7 , 5) and v- 3 w = (- 1 ,- 5) and c v + d w = (2 c + d,c + 2 d ). 5 u + v = (- 2 , 3 , 1) and u + v + w = (0 , , 0) and 2 u +2 v + w = (add first answers) = (- 2 , 3 , 1). 6 The components of every c v + d w 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 2 v (or 2 w ). 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 , , 1) , (1 , 1 , 0) , (1 , , 1) , (0 , 1 , 1) , (1 , 1 , 1). The center point is ( 1 2 , 1 2 , 1 2 ). The centers of the six faces are ( 1 2 , 1 2 , 0) , ( 1 2 , 1 2 , 1) and (0 , 1 2 , 1 2 ) , (1 , 1 2 , 1 2 ) and ( 1 2 , , 1 2 ) , ( 1 2 , 1 , 1 2 ). 12 A four-dimensional cube has 2 4 = 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 π 3 , sin π 3 ) = ( 1 2 , √ 3 2 ). 14 Sum = 12 j since j = (0 , 1) is added to every vector. 15 The point 3 4 v + 1 4 w is three-fourths of the way to v starting from w . The vector 1 4 v + 1 4 w is halfway to u = 1 2 v + 1 2 w , and the vector v + w is 2 u (the far corner of the parallelogram). 16 All combinations with c + d = 1 are on the line through v and w . The point V =- v + 2 w is on that line beyond w . 17 The vectors c v + c w fill out the line passing through (0 , 0) and u = 1 2 v + 1 2 w . It continues beyond v + w and (0 , 0). With c ≥ 0, half this line is removed and the “ray” starts at (0 , 0). 18 The combinations with 0 ≤ c ≤ 1 and 0 ≤ d ≤ 1 fill the parallelogram with sides v and w . 19 With c ≥ 0 and d ≥ 0 we get the “cone” or “wedge” between v and w . 20 (a) 1 3 u + 1 3 v + 1 3 w is the center of the triangle between u , v and w ; 1 2 u + 1 2 w is the center of the edge between u and w (b) To fill in the triangle keep c ≥ 0, d ≥ 0, e ≥ 0, and c + d + e = 1. 3 4 21 The sum is ( v- u ) + ( w- v ) + ( u- w ) = zero vector. 22 The vector 1 2 ( u + v + w ) is outside the pyramid because c + d + e = 1 2 + 1 2 + 1 2 > 1. 23 All vectors are combinations of u , v , and w . 24 Vectors c v are in both planes. 25 (a) Choose u = v = w = any nonzero vector (b) Choose u and v in different directions, and w to be a combination like u + v ....

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