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 (antisymmetric 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 offdiagonal 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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 Spring '12
 Minki
 Linear Algebra, Matrices, Dot Product, Mass, Diagonal matrix, Row

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