Introduction to Linear algebra-Strang-Solutions-Manual_ver13

A one possibility the matrices ca form a subspace not

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Unformatted text preview: er from L 1 . So B 1 is southeast. Northwest B D PL times southeast P U is .PLP /U D upper triangular. 37 Certainly B T is northwest. B 2 is a full matrix! B 1 38 There are nŠ permutation matrices of order n. Eventually two powers of P must be the same: If P r D P s then P r s D I . Certainly r s  n! "     01 P2 01 and P3 D 0 0 PD is 5 by 5 with P2 D 10 P3 10 # 0 1 and P 6 D I . 0 39 To split A into (symmetric B ) C (anti-symmetric C ), the only choice is B D and C D 1 .A 2 T T A /. 40 Start from Q Q D I , as in " # qT  1 qT 2 q1 q2   1 D 0 0 1  (a) The diagonal entries give q T q 1 D 1 and q T q 2 D 1: unit vectors 1 2 (b) The off-diagonal entry is q T q 2 D 0 (and in general q T q j D 0) 1 i   cos  sin  (c) The leading example for Q is the rotation matrix . sin  cos  1 .A CAT / 2 Solutions to Exercises 27 Problem Set 3.1, page 127 1 x C y ¤ y C x and x C .y C z/ ¤ .x C y / C z and .c1 C c2 /x ¤ c1 x C c2 x . 2 When c.x1 ; x2 / D .cx1 ; 0/, the only broken rule is 1 times x equals x . Rules (1)-(4) 3 4 5 6 7 8 9 10 11 for addition x C y still hold since addition is not changed. (a) cx may not be in our set: not closed under multiplication. Also no 0 and no x (b) c.x C y / is the usual .xy/c , while c x C c y is the usual .x c /.y c /. Those are equal. With c D 3, x D 2, y D 1 this is 3.2 C 1/ D 8. The zero vector is the number 1.       00 1 1 1 22 The zero vector in matrix space M is I AD and A D . 1 1 22 00 2 The smallest subspace of M containing the matrix A consists of all matrices cA. (a) One possibility: The matrices cA form a subspace not containing B (b) Yes: the subspace must contain A B D I (c) Matrices whose main diagonal is all zero. When f .x/ D x 2 and g .x/ D 5x , the combination 3f 4g in function space is h.x/ D 3f .x/ 4g .x/ D 3x 2 20x . Rule 8 is broken: If c f .x/ is defined to be the usual f .cx/ then .c1 C c2 /f D f ..c1 C c2 /x/ is not generally the same as c1 f C c2 f D f .c1 x/ C f .c2 x/. If .f C g /.x/ is the usual f .g .x// then .g C f /x is g .f .x// which is different. In Rule 2 both sides are f .g .h.x///. Rule 4 is broken there might be no inverse function f 1 .x/ such that f .f 1 .x// D x . If the inverse function exists it will be the vector f. (a) The vectors with integer components allow addition, but not multiplication by 1 2 (b) Remove the x axis from the xy plane (but leave the origin). Multiplication by any c is allowed but not all vector additions. The only subspaces are (a) the plane with b1 D b2 (d) the linear combinations of v and w (e) the plane with b1 C b2 C b3 D 0.     ab aa (a) All matrices (b) All matrices (c) All diagonal matrices. 00 00 12 For the plane x C y 13 14 15 16 17 2z D 4, the sum of .4; 0; 0/ and .0; 4; 0/ is not on the plane. (The key is that this plane does not go through .0; 0; 0/.) The parallel plane P0 has the equation x C y 2z D 0. Pick two points, for example .2; 0; 1/ and .0; 2; 1/, and their sum .2; 2; 2/ is in P0 . (a) The subspaces of R2 are...
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