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Math 33a/2, Quiz 4bd, November 8, 2007
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1. Find an orthonormal basis for the subspace
V
of
R
4
, where
V
= ker
±
1 0 4

3
0 1 2

1
²
.
Solution.
The matrix
±
1 0 4

3
0 1 2

1
²
is already in rref, so we can easily write down a
basis for its kernel:
~v
1
=

4

2
1
0
,
~v
2
=
3
1
0
1
.
To ﬁnd an orthonormal basis
{
~u
1
, ~u
2
}
for this space, we apply the GramSchmidt process.
~u
1
=
~v
1

~v
1

=
1
√
21

4

2
1
0
.
To obtain
~u
2
, we take
~v
2

(
~v
2
·
~u
1
)
~u
1
and divide by its length.
~v
2

(
~v
2
·
~u
1
) =
3
1
0
1

(
3
1
0
1
·
1
√
21

4

2
1
0
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This note was uploaded on 04/02/2008 for the course MATH 33a taught by Professor Lee during the Fall '08 term at UCLA.
 Fall '08
 lee
 Math

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