Math 103B Homework Solutions
HW5
Jacek Nowacki
May 13, 2007
Chapter 19
Problem 2.
Prove that a nonempty subset
U
of a vector space
V
over a field
F
is a subspace of
V
if, for every
u
and
u
in
U
and every
a
in
F
,
u
+
u
∈
U
and
au
∈
U
.
Proof.
Remark:
Let’s begin by noting that if you are investigating a subset
defined in some universal way (e.g. a kernel of a homomorphism), then it is
necessary to check that the subset is nonempty. It does not make sense to
define something and then see that it does not exist, i.e. get an empty set.
OK, we begin the proof. It is implied in the problem that the operation
in
U
is inherited from
V
.
Hence, we do not need to check that the four
conditions are satisfied – they are automatically true. A subspace is just a
subgroup (under addition) that is also closed under scalar multiplication.
Now, one of our subgroup tests tells us that all we have to do is show
that for all
x, y
∈
U
(we can do this since
U
is not empty),
x

y
∈
U
. It
should be clear that

y
= (

1)
·
y
, so that

y
is in
U
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 Winter '11
 PhanThuongCang
 Linear Algebra, Vector Space, ·, Jacek Nowacki, 1 spans

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