Math 110
First Midterm Examination
February 16, 2010
2:10–3:30 PM, 10 Evans Hall
Please put away all books, calculators, and other portable electronic devices—
anything with an ON/OFF switch.
You may refer to a single 2sided sheet
of notes. For numerical questions,
show your work
but do not worry about
simplifying answers. For proofs, write your arguments in complete sentences
that explain what you are doing. Remember that your paper becomes your only
representative after the exam is over.
Problem
Your score
Possible points
1
5 points
2
12 points
3
6 points
4
7 points
Total:
30 points
1.
In
R
3
, express (3
,
18
,

11) as a linear combination of (1
,
2
,
3), (

2
,
0
,
3) and (2
,
4
,
1).
This was a standard numerical problem of the type that most of you know how to do. The
coeﬃcients are:

49
/
5, 3 and 47
/
5. I apologize for the fractions: I intended the answers to
be whole numbers and must have mistyped.
2.
Label each of the following statements as TRUE or FALSE. Along with your answer,
provide an informal proof or an explanation. For false statements, an explicit counterexample
might work best. In interpreting the statements, take
v
to be a vector,
a
to be a scalar,
β
to be a basis of
V
, etc., etc.
Each part was worth 2 points. We gave out one point for the correct T/F answer and one
point for the explanation.
a.
If
av
=
v
, then either
a
= 1 or
v
= 0.
This is true, but a lot of you didn’t give a good reason. Since
v
= 1
·
v
, as proved in class,
the equation
av
=
v
may be written (
a

1)
·
v
= 0. If the scalar
a

1 is nonzero, we may