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MATH 110  SOLUTION TO QUIZ 1
LECTURE 1, SUMMER 2009
GSI: SANTIAGO CA
˜
NEZ
Let
V
be a vector space over a ﬁeld
F
, and let
U
and
W
be subspaces of
V
. Prove
that
U
∩
W
is a subspace of
V
.
Proof.
First, since
U
and
W
are both subspaces of
V
, we know that 0
∈
U
and 0
∈
W
.
Hence 0
∈
U
∩
W
. Now, let
x,y
∈
U
∩
W
. Then
x,y
∈
U
and
x,y
∈
W
. Since
U
is
closed under addition, we have that
x
+
y
∈
U
, and similarly since
W
is closed under
addition,
x
+
y
∈
W
. Thus
x
+
y
∈
U
∩
W
so
U
∩
W
is closed under addition.
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This note was uploaded on 08/31/2009 for the course MATH 110 taught by Professor Gurevitch during the Summer '08 term at University of California, Berkeley.
 Summer '08
 GUREVITCH
 Math, Vector Space

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