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MATH 223, Linear Algebra
Winter, 2008
Solutions to Assignment 4
1. Let
V
=
Z
4
2
and
W
1
=
Span
1
0
1
0
,
0
1
0
1
and
W
2
=
Span
0
1
1
0
,
1
0
0
1
be subspaces of
V
. Find a basis for
W
1
+
W
2
and one for
W
1
∩
W
2
.
Solution: Note that clearly
W
1
and
W
2
are both 2dimensional, so the sets
that are given to span them are bases for them. Considering the matrix
1 0

0 1
0 1

1 0
1 0

1 0
0 1

0 1
, the column space of which is
W
1
+
W
2
, and the fact
that it rowreduces very quickly to
1 0

0 1
0 1

0 1
0 0

1 1
0 0

0 0
, it’s immediate
that a basis for
W
1
+
W
2
is
1
0
1
0
,
0
1
0
1
,
0
1
1
0
.
For the inter
section, the RREF tells us that that the fourth column is the sum of the
ﬁrst three, so the sum of the ﬁrst two equals the sum of the last two,
which is a nonzero vector in the intersection. Also Lunch In Chinatown
tells us that the intersection has dimension 1, so a basis for
W
1
∩
W
2
is
just
1
1
1
1
.
2. Let
V
=
C
5
and
W
1
=
Span
1
i
1
i
1
,
2
i

1
1 +
i
0
2
,
4
3
i
3

i
2
i
2

2
i
,
2 + 2
i

1 + 2
i
3 +
i
1 + 2
i
4
and
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 Fall '07
 Loveys
 Linear Algebra, Algebra

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