MATH1111/Linear Algebra/25Oct10/Test 2/a
1
THE UNIVERSITY OF HONG KONG
DEPARTMENT OF MATHEMATICS
Test 2: Suggested solution
±
The set
V
of 2
²
2 nonsingular matrices
±
a b
c d
²
under the usual matrix
operations is a vector space.
F
Explanation
: Clearly
V
³
R
2
±
2
. If
V
is a vector space w.r.t. the usual matrix operations,
then
V
is a subspace of
R
2
±
2
. So the zero matrix 0
2
V
. The subspaces of a vector space
share the same zero element. However, 0 is singular! (Compare with Tutorial II Qn 3
(b).)
±
The set
V
0
of 2
²
2 matrices
±
a b
c d
²
where
b
+
c
= 0 under the usual matrix
operations is a vector space.
T
Explanation
: You may give a proof in a way similar to Tutorial II Qn 3 (c). Alternatively,
observe that it su±ces to show
V
0
is subspace of
R
2
±
2
.
±
Let
x
and
y
be vectors in a vector space. If
x
6
=
y
, then Span(
x
) + Span(
y
)
is a subspace of dimension 2.
F
Explanation
: Consider the vector space
R
2
,
x
=
±
1
0
²
and
y
= 2
x
. Then Span(
x
) +
Span(
y
) = Span(
x
), which is a subspace of dimension 1.
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 Spring '11

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