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110.201 Linear Algebra
4th Quiz
April 8, 2005
Problem 1
Let
~e
i
,
i
= 1
,
2
,
3
,
4 be the standard basis of
R
4
. Is there an
orthogonal matrix
A
with
A~e
1
=
1
√
2
0
0
1
√
2
, A~e
2
=
1
√
2
0
0

1
√
2
, A~e
3
=
0
1
0
0
, A~e
4
=
0
0
2
0
or not? If so, ﬁnd
A
, and if not, explain.
Problem 2
Find an orthonormal basis for the subspace
V
⊂
R
3
spanned by
the vectors
1
0
1
,
1
1
0
.
Moreover, ﬁnd a normal basis for the orthogonal complement of
V
.
Problem 3
Let
{
~α
1
, ~α
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Unformatted text preview: 2 ,..., ~Î± k } be a set of mutually orthogonal vectors in R n . 1. Show that for any ~v âˆˆ R n , the vector ~v(proj ~Î± 1 ( ~v ) + proj ~Î± 2 ( ~v ) + Â·Â·Â· + proj ~Î± k ( ~v )) is orthogonal to each of the ~Î± 1 , ~Î± 2 ,..., ~Î± k . 2. Verify this claim for R 3 with ~ Î± 1 = ~ e 1 , ~ Î± 2 = ~ e 2 , and letting ~v = 1 2 3 . Draw a picture. 1...
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This homework help was uploaded on 01/23/2008 for the course MATH 201 taught by Professor Consani during the Spring '05 term at Johns Hopkins.
 Spring '05
 CONSANI
 Linear Algebra, Algebra

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