Math 167 homework 2 solutions
October 13, 2010
1.5.44: Find a 3 by 3 permutation matrix with P 3 = I
Find a 4 by 4 permutation matrix with P with P 4 = I .
0010
001
1 0 0 0
P = 1 0 0
P =
0 1 0 0
010
Math 167 homework 5 solutions
November 4, 2010
3.3.4: Write out E 2 = |Ax - b|2 and set to zero its derivatives with respect to u and v, if 1 0 1 0 1 , x = u , b = 3 . A= v 1 1 4 Compare the resulting
Math 167 homework 8 solutions
November 27, 2010
5.3.2: Bernadelli studied a beetle "which lives three years only, and propagates in its third year." They survive the first year with probability 1 , an
Solutions to Exercises
150
Problem Set 9.1, page 436
1 (a)(b)(c) have sums 4, 2 + 2i, 2 cos and products 5, 2i, 1. Note (ei )(ei ) = 1.
2 In polar form these are
i
5e , 5e2i ,
3 The absolute values a
Solutions to Exercises
150
Problem Set 9.1, page 436
1 (a)(b)(c) have sums 4, 2 + 2i, 2 cos and products 5, 2i, 1. Note (ei )(ei ) = 1.
2 In polar form these are
i
5e , 5e2i ,
3 The absolute values a
Solutions to Exercises
150
Problem Set 9.1, page 436
1 (a)(b)(c) have sums 4, 2 + 2i, 2 cos and products 5, 2i, 1. Note (ei )(ei ) = 1.
2 In polar form these are
i
5e , 5e2i ,
3 The absolute values a
Solutions to Exercises
155
10 For every integer n, the nth roots of 1 add to zero. For even n, they cancel in pairs. For
any n, use the geometric series formula 1 + w + + wn1 = (wn 1)/(w 1) = 0.
In pa
Solutions to Exercises
150
Problem Set 9.1, page 436
1 (a)(b)(c) have sums 4, 2 + 2i, 2 cos and products 5, 2i, 1. Note (ei )(ei ) = 1.
2 In polar form these are
i
5e , 5e2i ,
3 The absolute values a
Solutions to Exercises
16 r = 1, angle
2
151
; multiply by ei to get ei/2 = i.
17 a + ib = 1, i, 1, i, 12
i .
2
The root w = w1 = e2i/8 is 1/ 2 i/ 2.
18 1, e2i/3 , e4i/3 are cube roots of 1. The cub
Solutions to Exercises
85
Problem Set 5.1, page 254
1 det(2A) = 24 det A = 8; det(A) = (1)4 det A = 12 ; det(A2 ) = 14 ; det(A1 ) = 2.
2 det( 12 A) = ( 12 )3 det A = 81 and det(A) = (1)3 det A = 1; de
Math 167 homework 7 solutions
November 17, 2010
5.1.12: Find the eigenvalues and eigenvectors of A= 3 4 4 -3 and B = a b b a .
The characteristic polynomial for A is p() = (3 - )(-3 - ) - 16 = 2 - 25.
Math 167 homework 6 solutions
November 11, 2010
3.4.2: Project 0 b = 3 0
If follows that
and then find its projection p onto the plane of a1 and a2 . Since a1 and a2 are orthonormal vectors, the matri
Math 167 homework 4 solutions
October 31, 2010
2.4.2: Find the dimension and construct a basis for the four subspaces associated with each of the matrices A= Matrix A: 1. dim C(A) = 1, basis is 1 2 0
Math 167 homework 3 solutions
October 20, 2010
2.1.22: For which right-hand sides (nd a condition on b1 , b2 , b3 ) are these
systems solvable?
(a)
1
4
2
x1
b1
2
8
4 x2 = b2
1 4 2
x3
b3
r2 2r1 and r3
Math 167 homework 1 solutions
October 6, 2010
1.3.14a: Construct a 3 by 3 system that needs two row exchanges to reach
a triangular form and a solution. There are many examples, but this is one:
3y 2z