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
0001
1.6.2
(but not P = I ).
(a) Find the inverses of t
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 , and 2 the second with probability 1 , and then produce si
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 = 12
z
= 3
4x +3y
=3
1.3.14b: Construct a 3 by 3 syste
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 + r1 give the conditions b2 2b1 = 0 and b3 + b1 = 0 fo
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 4 , -1 0 0 1 4 0 0 2 8 0 and U = 0 1 4 0 0 0 0 0
0 1 0
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 equations with AT A^ = AT b, confirming that calculus
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 matrix P of the projection onto the subspace spanned by a1 a
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. Thus the eigenvalues of A are 1 = 5 and 2 = -5; corres