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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
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This note was uploaded on 05/04/2010 for the course MATH 136 taught by Professor All during the Spring '08 term at Waterloo.
 Spring '08
 All
 Linear Algebra, Algebra, Vector Space

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