University of Toronto at Scarborough
Department of Computer and Mathematical Sciences
Linear Programming and Optimazation
MATB61 Winter 2008
Solution set to Assignment 2
Addition:
1.
a) Let
⎥
⎦
⎤
⎢
⎣
⎡
=
⎥
⎦
⎤
⎢
⎣
⎡
+
⎥
⎦
⎤
⎢
⎣
⎡
+
⎥
⎦
⎤
⎢
⎣
⎡
+
⎥
⎦
⎤
⎢
⎣
⎡
0
0
0
0
0
0
0
1
1
0
1
1
0
1
1
0
1
0
0
1
4
3
2
1
r
r
r
r
where
r
i
∈
R, i = 1, 2, 3, 4.
Then
⎥
⎦
⎤
⎢
⎣
⎡
=
⎥
⎦
⎤
⎢
⎣
⎡
+
+
+
+
0
0
0
0
3
1
2
3
2
4
3
1
r
r
r
r
r
r
r
r
Solve the equations:
r
1
+
r
2
+
r
4
= 0
r
2
+
r
3
= 0
r
2
= 0
r
1
+
r
3
= 0
It follows that
r
1
=
r
2
=
r
3
=
r
4
= 0. Therefore the set is independent.
b) Let
r
1
(1 + x) +
r
2
(x + x
2
) +
r
3
(x
2
+ x
3
) +
r
4
x
3
= 0
where
r
i
∈
R, i = 1, 2, 3, 4.
Rearrange the equation,
r
1
+(
r
1
+
r
2
)x
+
(
r
2
+
r
3
)x
2
+ (
r
3
+
r
4
)x
3
= 0.
Since {1, x, x
2
, x
3
} is independent, we have
r
1
= 0
r
1
+
r
2
= 0
r
2
+
r
3
= 0
r
3
+
r
4
= 0
It follows that
r
1
=
r
2
=
r
3
=
r
4
= 0. Therefore the set is independent.
2.
a) Since {p(x)  p(x)= p( – x)}
P
2
, Let p(x) = ax
2
+ bx + c.
⊂
If p(x)
∈
{p(x)  p(x)= p( – x)}, we have
ax
2
+ bx + c = a( x)
2
+ b( x) + c = ax
2
– bx + c.
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 Spring '10
 X.Jiang
 Linear Algebra, Algebra, Linear Programming, Addition, SEPTA Regional Rail, Jaguar Racing, Scarborough Department of Computer and Mathematical Sciences, r2 + r3, Optimazation

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