MAT 1341A: These are the solutions to the
practice assignment.
Don’t read these until
you have attempted the questions on your own!
Question # 1:
(a)
Determine whether
4
8
0
1
∈
span
2
1

3
5
,
0
2

1
3
⊆
R
4
.
Solution:
We need to try to solve:
4
8
0
1
=
a
2
1

3
5
+
b
0
2

1
3
for some
a, b
∈
R
. This gives the following 4 linear equations:
2
a
= 4
,
a
+ 2
b
= 8
,

3
a

b
= 0
,
5
a
+ 3
b
= 1
The first gives
a
= 2, so the second implies
b
= 3.
But plugging this into the third equation gives a
contradiction. So we conclude this system cannot be solved. Hence, our answer is NO.
(b)
Determine whether
10
6

8
22
∈
span
1
2
0
3
,
1
0

2
1
⊆
M
2
,
2
.
Solution:
We need to try to solve:
10
6

8
22
=
a
1
2
0
3
+
b
1
0

2
1
for
a, b
∈
R
. Equating terms in each component gives the four equations:
a
+
b
= 10
,
2
a
= 6
,

2
b
=

8
,
3
a
+
b
= 22
The second gives
a
= 3 and the third gives
b
= 4, which fails the first equation, so there is no solution, and
again our answer is NO.
Question # 2:
Let
P
1
be the vector space of all polynomials with degree
≤
1. Show that any element of
P
1
is in the span of the polynomials
x

2, 2
x
+ 1, and 2. (
Hint: take an arbitrary element of
P
1
, which looks
like
ax
+
b
for
a, b
∈
R
, and see if you can write it as a linear combination of these vectors.
)
Solution:
We need to decide if, for any
ax
+
b
in
P
1
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 Winter '10
 DAIGLE
 Vectors, Vector Space

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