1.2: 18
18. Nitric acid is prepared commercially by a series of three chemical reactions.
In the rst reaction, nitrogen (N2 ) is combined with hydrogen (H2 ) to form
ammonia (NH3 ). Next the ammonia is combined with oxygen (O2 ) to form
nitrogen dioxide (
4.3: 11, 12, 14
11. Show that if A and B are similar matrices then det(A) = det(B ).
Proof. Suppose A and B are similar matrices. Then B = S 1 AS for some
nonsingular matrix S . Since S is nonsingular, we know that det(S ) = 0 and
1
det(S 1 ) = det(S ) .
Math 302 homework solutions for Sec 6.1
4. Let A be a nonsingular matrix and let be an eigenvalue of A. Show that
A1
1
is an eigenvalue of
Proof. Let x be the eigenvector corresponding to . Then
Ax = x
(1)
multuply both sides by A1 and devide both sides b
43
2.1. THE DOT PRODUCT
2.1.6
Work
An important application of the dot product is the concept of work. The physical concept
of work does not in any way correspond to the notion of work employed in ordinary conversation. For example, if you were to slide a
Math 302 Test 1 Review
1. Given two points in R3 , (x1 , y1 , z1 ) and (x2 , y2 , z2 ) , show the point
x1 + x2 y1 + y2 z1 + z2
,
,
2
2
2
is on the line between these two points and is the same distance from each of them.
Answer: The line is r(t) = x1 , y
A Sample Of Test Questions
1
Find the differential df at the point x, y 0, 1 given that f x, y 3y x 2 y
2y sin x.
3y x 2 y 2y sin x The gradient is 2xy 2y cos x, x 2 2 sin x 3 Then at the point of interest, the
gradient is 2, 3 . Then the differential is
Test 4 Review
1. The ice cream in a sugar cone occupies the region bounded between the sphere x2 + y 2 + z 2 = 16 and the cone
z = 3x2 + 3y 2 . If the units are in inches, nd the total volume of this ice cream in cubic inches using:
(a) cylindrical coordi
A Concatenation Of Diabolical Terminology and
Nefarious Denitions
gray area! Thus the set of vectors, cfw_u1 , u2 , , um is
independent if whenever
n
1. Linear combination.
ci ui = 0
Let cfw_u1 , u2 , , um be some vectors. A linear combination is someth
Sample Problems For Test 1
1
1
Let A
2
21
matrix equation XA B X?
X A I B and so X B A
5
2
X
1
I
5
2
and B
2
. Which 2
2 matrix X is the solution to the
1
3
10
2
Find the cosine of the angle between the two vectors
1
102
102
2, 2, 3 and
2, 1, 1 .
3
Find
Math 302 homework solutions for Sec 4.1
6. Determine whether the following are linear transformations from R2 into R3
(a) L(x) = (x1 , x2 , 1)t
(b) L(x) = (x1 , 0, 0)T
Proof. Let x1 , X2 be two vectors in R2 and , be scalars, then
(a) L(x) = (x, y, 1)t
L(
3.6: 1c, 11, 15
1c. Find a basis for the row space, a basis for the column space, and a basis for
the nullspace of the matrix
1 3 2 1
A = 2 1 3 2 .
34 5 6
Answer. The reduced row echelon
1
B = 0
0
form of A is
0 0 13/20
1 0 21/20 .
01
3/4
Thus the row vec
Math 302 homework solutions for Sec 1.3
26 Let A be an m n matrix.
1. Explain why the matrix nultiplications AT A, and AAT are possible.
2. Show that AT A and AAT are both symmetric.
Proof.
1. Because both AT is an n m matrix, so the dimension matches.
2.
1.4: 12c, 18, 22
12c. Given
5
3
A=
3
,
2
6
2
B=
2
,
4
solve the following matrix equation:
AX + B = X.
Answer. If A I is nonsingular, then we can solve the system in the following
algebraic manner:
AX + B
AX X
AX IX
(A I )X
1
(A I ) (A I )X
IX
X
=
=
=
=
=
Math 302 homework solutions for Sec 2.1
9. Prove that if a row and a column of an n n matrix A consists entriely of zeros then det(A)
=0.
Proof. This can be proved really easily by Theorem 2.1.1. Just choose the zero row or the zero
column for the cofacto
2.2: 6, 15
6. Let A be a nonsingular matrix. Show that
det(A1 ) =
1
.
det(A)
Answer. Recall that for n n matrices A and B , det(A) det(B ) = det(AB ) by
Theorem 2.2.3. Thus
det(A1 ) det(A) = det(A1 A) = det(I ) = 1.
But det(A1 ) and det(A) are scalars, so
Math 302 homework solutions for Sec 2.3
Exercise 8. Let A be a nonsingular n n matrix with n > 1. Show that
det(adjA) = (det(A)n1
Proof. Note that
A 1 =
adjA
det(A)
then
(det(A)1 = det(A1 ) = det
thus we got the result.
1
adjA
det(A)
=
det(adjA)
(det(A)n
3.1: 12
12. Let R+ denote the set of positive real numbers. Dene the operation of
scalar multiplication, denoted , by
x = x
for each x R+ and for any real number . Dene the operation of addition,
denoted , by
xy =xy
for all
x, y R+ .
Thus for this system
3.1: 11, 16, 17
11. Prove that any nite set of vectors that contains the zero vector must be
linearly dependent.
Proof. If cfw_v1 , v2 , . . . , vn is a nite set of vectors and one of them is the zero
vector, without loss of generality we may assume that
Math 302 homework solutions for Sec 3.4
7. Find a basis for the subspace S of R4 consisting of all vectors of the form (a + b, a b + 2c, b, c)T ,
where a, b, and c are real numbers. What is the dmimension of S ?
Proof. the basis is
1
1
b ,
1 0
0
1
1
Test 3 Review
1. Prove that the direction of maximal rate increase of a function f (x, y ) is in the direction of f and that the maximal
rate of decrease is in the direction of - f . What are the rates of increase in these directions?
2. Find the normal a