Practice Problems
2/13/06
(1) Let
u
and
v
be two distinct vectors of a vector space
V
. Let
{
u, v
}
be a basis for
V
and
a, b
nonzero scalars. Show that
{
u
+
v, au
}
and
{
au, bv
}
are also bases for
V
.
(2) The set of all (3
×
3) matrices having trace equal to zero is a subspace
W
of
M
3
×
3
. Find a basis for
W
.
Solution: Note that any (3
×
3) matrix
a
b
c
,
can be written
as a linear combination of
0
1
0
0
0
0
0
0
0
,
0
0
1
0
0
0
0
0
0
,
0
0
0
1
0
0
0
0
0
,
0
0
0
0
0
1
0
0
0
,
0
0
0
0
0
0
1
0
0
,
0
0
0
0
0
0
0
1
0
,
1
0
0
0

1
0
0
0
0
,
0
0
0
0

1
0
0
0
1
. This is a linearly independent, generating set. Hence
the dimension of
W
is 8.
(3) The set of all (3
×
3) upper triangular matrices is a subspace
W
of
M
3
×
3
. Find a basis for
W
.
(4) Let
V
be a vector space having dimension
n
, and let
S
be a subset
of
V
that generates
V
. Prove that there is a subset of
S
that is a
basis for
V
. (Do not assume
S
to be finite.)
(5) Let
W
1
, W
2
be subspaces of a finite dimensional vector space
V
.
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 Spring '10
 SHALOM
 Linear Algebra, Algebra, Vectors, Matrices, Scalar, Vector Space

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