MA 265 HOMEWORK ASSIGNMENT #4 SOLUTIONS
#1. Page 114; Exercise 6. Repeat Exercice 5 for the following linear systems: (a) x x 3x x x 2x 2x 3x x 6x 5x + y - 2y + y + y + y + y + y - 2y + y - y + 2z + z + z + z - 2z + z + + - + + z z z z 2z + 3w + w - w = 1
Quiz 4
MA 262
Arturs Class
February 14, 2012
Problem 1
Put
B=
01
10
, C=
10
0 1
.
Compute the commutator [B, C ] := BC CB .
Problem 2
Put
A=
3 1
5 1
.
Compute A2 .
Problem 3
With A as above, what is A2 A A A2 ?
Problem 4
Compute the reduced row echelon fr
Quiz 3
MA 262
Arturs Class
February 5, 2012
Problem 1
Give all values of n, c such that
f (x, y ) = xn + cy 2
is homogeneous.
Problem 2
Give all values such that
dy + 3xy (4 + y ) dx = 0
is
(i) Bernoulli,
(ii) Linear.
DO NOT SOLVE.
Problem 3
Is
2xy dx + (
Quiz 2
MA 262
Arturs Class
January 24, 2012
Problem 1
Find a nonconstant solution y = y (x) of
dy
2
= (y 1)1/2
dx
3
satisfying y (1) = 1. This solution is dened over which domain?
Problem 2
Use an integrating factor to nd a solution y = y (x) of
(y ex ) d
Quiz 1
MA 262
Arturs Class
January 18, 2012
Problem 1
You are on a dierent planet. Objects dropped at 1 meter above the surface strike
the ground 1 second later. Ignoring atmospheric resistance, etc., set up and solve
the initial value problem (IVP) and r
Quiz 6 Solutions
MA 262
Arturs Class
February 29, 2012
Problem 1
A=
20
00
Compute nullspace(A).
Solution
Recall that the nullspace of A is the set of vectors v R2 that are mapped to
0 under A, i.e., Av = 0. Suppose v = (x1 , x2 ) is in the nullspace of A.
Quiz 5
MA 262
Arturs Class
February 21, 2012
Problem 1
Put
1
0
A=
0
0
2
5
0
0
3
6
8
0
4
7
.
9
1
Compute det(A)
Solution
This matrix is upper triangular (UT). So we can compute along the diagonal:
det(A) = 1 5 8 1 = 40.
(This is an immediate consequence of
Quiz 4
MA 262
Arturs Class
February 14, 2012
Problem 1
Put
B=
01
10
, C=
10
0 1
.
Compute the commutator [B, C ] := BC CB .
Solution
First compute both products
BC =
0 1
10
01
1 0
, CB =
So their dierence is
0 +2
2 0
BC CB =
.
Problem 2
Put
A=
3 1
5 1
.
C
Quiz 2 Solutions
MA 262
Arturs Class
February 13, 2012
Problem 1
Find a nonconstant solution y = y (x) of
dy
2
= (y 1)1/2
dx
3
satisfying y (1) = 1. This solution is dened over which domain?
Solution
Rearrange and apply integrals to get
1
2
1
dy
=
1/2
(y
Quiz 3 Solutions
MA 262
Arturs Class
February 13, 2012
Problem 1
Give all values of n, c such that
f (x, y ) = xn + cy 2
is homogeneous.
Solution
A function f of two variables x, y is homogeneous when f (tx, ty ) = tk f (x, y ),
where k is the degree of h
Quiz 1
MA 262
Arturs Class
January 18, 2012
Problem 1
You are on a dierent planet. Objects dropped at 1 meter above the surface strike
the ground 1 second later. Ignoring atmospheric resistance, etc., set up and solve
the initial value problem (IVP) and r
MA 262
Practice for Exam I
1. If y is the solution to the equation
y=
x2 + 1
,
y3
y (0) = 2,
then y (3) =
A. 2
B. 2 2
C. 4 2
D. 0
E. 4
2. If y is the solution to the equation
y=
3
y + x,
x
y (1) = 0,
then y ( 3) =
A. 3 3
B. 3 3
C. 3( 3 1)
D. 3( 3 + 1)
27
Quiz 5
MA 262
Arturs Class
February 21, 2012
Problem 1
Put
1
0
A=
0
0
2
5
0
0
3
6
8
0
4
7
.
9
1
Compute det(A)
Problem 2
Recall C (R) is the real vector space of continuous functions on R. The polynomials
of degree 2 form a subset. Show that this is also
Quiz 6
MA 262
Arturs Class
February 28, 2012
Problem 1
A=
20
00
A=
00
00
A=
20
01
Compute nullspace(A).
Problem 2
Compute nullspace(A).
Problem 3
Compute nullspace(A).
Problem 4
Consider the dierential equation
y + 2y y = 1.
(a) Write down the solution sp
MA 265 HOMEWORK ASSIGNMENT #1 SOLUTIONS
#1. Page 8; Exercise 1. In Exercises 1 through 14, solve each given linear system by the method of elimination. x + 2y = 8 3x - 4y = 4
Solution: First we eliminate x. Multiply the first equation by (-3), then add to
MA 265 GOINS MIDTERM EXAMINATION #2
Name:
Instructions:
Circle the correct answer on the following pages. You have 50 minutes to complete 25 problems. No textbooks, personal notes, calculators, or computing aids are allowed during the examination period.
MA 265 GOINS MIDTERM EXAMINATION #1
Name:
Instructions:
Circle the correct answer on the following pages. You have 50 minutes to complete 35 problems. No textbooks, personal notes, calculators, or computing aids are allowed during the examination period.
MA 265 HOMEWORK ASSIGNMENT #10 SOLUTIONS
#1. Page 297; Exercise 2. In Exercises 1 and 2, find the length of each vector. 0 (a) -2 0 -1 (b) -3 -4 1 (c) -2 4
Solution: (a) We define the 3-vector 0 v = -2 0 (b) We define the 3-vector -1 v = -3 -4 1 v = -2 4
MA 265 HOMEWORK ASSIGNMENT #9 SOLUTIONS
#1. Page 282; Exercise 1. Find a basis for the subspace V of R3 spanned by 2 -1 0 1 1 S = 2 , 1 , -1 , 1 , 1 3 4 2 2 1 and write each of the following vectors in terms of the basis vectors: 3 (a) 4 12 3 (b) 2 2 1 (c
MA 265 HOMEWORK ASSIGNMENT #7 SOLUTIONS
#1. Page 207; Exercise 17. (a) The set of all n n (b) The set of all n n (c) The set of all n n
Which of the following subsets of the vector space Mnn are subspaces? symmetric matrices diagonal matrices nonsingular
MA 265 HOMEWORK ASSIGNMENT #6 SOLUTIONS
2 #1. Page 169; Exercise 2. Let A = -1 3
1 3 2 0 . -2 1
(a) Find adj A. (b) Compute det(A). (c) Verify Theorem 3.12; that is, show that A (adj A) = (adj A) A = det(A) I3 .
Solution: (a) First we compute the cofacto
MA 265 HOMEWORK ASSIGNMENT #5 SOLUTIONS
#1. Page 145; Exercise 3. Determine whether each of the following permutations of S = cfw_1, 2, 3, 4 is even or odd: (a) 4 2 1 3 (b) 1 2 4 3 (c) 1 2 3 4
Solution: (a) This permutation sends 1 4 2 2 3 1 43
4213 :
Rea