HOMEWORK ASSIGNMENT 1 SOLUTIONS
§
1.1
#
3.b
Let
A
= (3
,

6
,
7),
B
= (

2
,
0
,

4), and
C
= (5
,

9
,

2) be the three given points.
Set
u
=
B

A
= (

2
,
0
,

4)

(3
,

6
,
7) = (

5
,
6
,

11)
and
v
=
C

A
= (5
,

9
,

2)

(3
,

6
,
7) = (2
,

3
,

9)
.
Following example 2 in the book, the equation for the plane is
x
=
A
+
su
+
tv
= (3
,

6
,
7) +
s
(

5
,
6
,

11) +
t
(2
,

3
,

9)
.
§
1.1
#
3.d
Let
A
= (1
,
1
,
1),
B
= (5
,
5
,
5), and
C
= (

6
,
4
,
2) be the three given points. Set
u
=
B

A
= (5
,
5
,
5)

(1
,
1
,
1) = (4
,
4
,
4)
and
v
=
C

A
= (

6
,
4
,
2)

(1
,
1
,
1) = (

7
,
3
,
1)
.
Again, following example 2 in the book, the equation for the plane is
x
=
A
+
su
+
tv
= (1
,
1
,
1) +
s
(4
,
4
,
4) +
t
(

7
,
3
,
1)
.
§
1.1
#
7
This was done in class.
§
1.2
#
1
(a) True. This is axiom (VS 3) of a vector space.
(b) False. The zero vector is unique by Corollary 1, page 11.
(c) False. For example, let
a
= 0,
b
= 1, and
x
=
~
0. Then
ax
=
bx
=
~
0, but
a
6
=
b
.
(d) False. For example, consider the vector space
R
2
. If
a
= 0,
x
= (1
,
1) and
y
= (1
,
0)
then
ax
=
ay
= (0
,
0) but
x
6
=
y
.
(e) True. Just think of them as column vectors.
(f) False. An
m
×
n
matrix has
m
rows and
n
columns.
(g) False. For example, we can add
x
+ 1 and
x
2
to get
x
2
+
x
+ 1.
1