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Math 136
Assignment 1 Solutions
1.
Determine whether
4

1
2
is in the span of
1
1

6
,
2

1
3
,
0
0
2
.
Solution: We need to determine whether there exists coeﬃcients
c
1
,
c
2
, and
c
2
such that
4

1
2
=
c
1
1
1

6
+
c
2
2

1
3
+
c
3
0
0
2
Using operations on vectors on the right hand side, we get
4

1
2
=
c
1
c
1

6
c
1
+
2
c
2

c
2
3
c
2
+
0
0
2
c
3
=
c
1
+ 2
c
2
c
1

c
2

6
c
1
+ 3
c
2
+ 2
c
3
Two vectors are equal if and only if there corresponding entries are equal. Thus, we have
a system of 3 equations in 3 unknowns:
c
1
+ 2
c
2
= 4
c
1

c
2
=

1

6
c
1
+ 3
c
2
+ 2
c
3
= 2
We can solve this using substitution and elimination. Subtracting the second equation
from the ﬁrst gives

3
c
2
=

5, so
c
2
= 5
/
3. Substituting this into the ﬁrst equation
we get
c
1
+ 2(5
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This note was uploaded on 07/16/2011 for the course MATH 136 taught by Professor All during the Fall '08 term at Waterloo.
 Fall '08
 All
 Math

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