Math
136
Assignment
6
Due:
Wednesday,
Mar
3rd
1.
Determine, with proof, which of
the
following are subspaces of the given vector space.
Find
a basis for each subspace.
a)
A
=
{(1':l,X2,X:3)
E]R3
12~rl
+X3
=
O,Xl
+X2
X3
=
o}
ofIR
3
.
b)
B
=
{ax2
+
b~r
+
c
E
P2
I
b
2

4ac
=
o}
of
P2·
c)
C
=
{ax
2
+
b:r
+
e
E
P2
I
a

c
=
b}
of
P
2 ·
d) D
=
{[~
~]
E
M(2,
2)
I
ad

be
=
o}
of
AI(2, 2).
e)
E
= {
[~
~]
E
Al
(2,
2)
I
a

b
+
e
=
° }
of ,\1 (2, 2).
2. Determine, with proof, whether
the
given set
is
a basis for
P
2
·
a)
{I
+
5x
+
x
2
,
2
+
:C:,
3
+
2
x
+
X2)
} .
b)
{I
+
2x
+
3x2,
3
+
2x
+
x
2
,
5
+
6x
+
7
x
2
}.
3. Invent a basis
for
each of the following subspaces of
A1(2,
2).
a)
The
set of all 2 x 2 lower
triangular
matrices.
b)
The
set of all 2 x 2 diagonal matrices.
4. Let
S
=
{VI,
...
,
'Un}
be a set of vectors
in
a vector space
1/'.
Prove
that
span
S
is
a
subspace of
1/.
5. Let
S
=
{(a,
b)
E
]R2
I
b
>
o}
and
define addition by
(a,
b)
+
(c,
d)
=
(ad
+
be,
bd)
and
define scalar multiplication by
k(a,
b)
=
(kab
k

l
,
b
k
).
Prove
that
S
is a vector space over
lR.
6. Let
1I
and
TV
be vector spaces
and
let
T
:
F
+
n
y
be a linear mapping.
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 Spring '08
 All
 Linear Algebra, Algebra, Vector Space, linearly dependent set

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