B U Department of Mathematics
Math 201 Matrix Theory
Spring 2005 Final Exam
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1.)
Consider the set of 2
×
2, symmetric matrices.
a)
Show that this is a subspace of all 2
×
2 matrices.
b)
Write a basis for this space and ﬁnd its dimension.
c)
Consider the set of
n
×
n
symmetric matrices. What is the dimension of this space? (You should
explain your answer.)
Solution:
a)
Let
S
2
×
2
denote the set of 2
×
2 symmetric matrices. Let
±
a
1
b
1
b
1
d
1
²
and
±
a
2
b
2
b
2
d
2
²
be
any two elements of
S
2
×
2
. Then, for
c
∈
R
, we have
±
a
1
b
1
b
1
d
1
²
+
c
±
a
2
b
2
b
2
d
2
²
=
±
a
1
+
ca
2
b
1
+
cb
2
b
1
+
cb
2
d
1
+
cd
2
²
∈
S
2
×
2
.
Hence
S
2
×
2
is a subspace of all 2
×
2 matrices.
b)
Clearly,
³±
1 0
0 0
²
,
±
0 0
0 1
²
,
±
0 1
1 0
²´
is a basis for
S
2
×
2
as its a linearly independent set
spanning
S
2
×
2
. Thus, dimension of a vector space being equal to the number of elements
in a basis, we get that dim
S
2
×
2
= 3.
c)
It is clear that to ﬁnd the number of basis elements of an
n
×
n
symmetric matrix we
need to count the number elements on the upper triangular part including the diagonal.
(This is because for a symmetric matrix elements below the diagonal are determined by
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 Fall '08
 SOYSAL
 Math, Linear Algebra, Matrices, Orthogonal matrix, linearly independent columns

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