MATH 223  HOMEWORK #1
Solutions
The problems in the book define vector spaces over a field
F
. You may assume
that we always have
F
=
R
. When solving the problems you may refer to any result
we have proved in class or that is proved in the book.
The numbers, such as 13 in Section 1.2, refer to the textbook. This first homework
has the full text of the problems copied from the book since some students don’t
have the textbook yet.
Problem
1. Let
V
be a vector space. Use the axioms to prove:
(i) For any vector
v
in
V
0
·
v
= 0
.
(Hint: add
v
+
w
= 0 to the left hand side and simplify, where
w
is the “negative”
of
v
from axiom (VS 4).)
(ii) For any vector
v
in
V
, the vector
w
= (

1)
·
v
satisfies axiom (VS 4):
v
+
w
= 0.
In other words, (

1)
·
v
is the “negative” of
v
.
(i) We can write:
0
·
v
= 0
·
v
+ 0 = 0
·
v
+
v
+
w.
where we used axiom (VS 3) in the first equality and
w
comes from axiom (VS
4). Now
v
= 1
·
v
by axiom (VS 5)and we can continue:
= 0
·
v
+ 1
·
v
+
w
= (0 + 1)
·
v
+
w
=
v
+
w
= 0
.
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 Spring '10
 SHALOM
 Linear Algebra, Algebra, Group Theory, Addition, Vector Space

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