Harvard University
Midterm 1 for Math 121, Fall 2007
Monday, October 22, 2007
Time allowed: 53 minutes
3 pages (including this one)
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Apart from question 1, you must fully justify your answers.
On this exam the full
score is 100 points.
You may assume all vector spaces are finitedimensional unless otherwise stated. Recall
that for a linear transformation
T
:
V
→
W
, the notation
N
(
T
) denotes the null space
and
R
(
T
) denotes the range.
Good Luck!
1
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Harvard University
Midterm 1 for Math 121, Fall 2007
Monday, October 22, 2007
1.
(20 points) Mark the following statements true or false. No justification is needed.
a)
Let
W
=
{
f
(
x
)
∈
P
(
R
)

f
(
x
) =
f
(

x
)
}
, where
P
(
R
) is the set of all polynomials
with real coefficients. (
P
(
R
) has infinite dimension.) Then
W
together with the
common addition and scalar multiplication is a vector space over
R
.
b)
The vectors (1
,
0
,
0)
,
(0
,
2
,
2)
,
(2
,
4
,
4) in
R
3
are linearly independent.
c)
Let
V
be a vector space.
Let
A
1
,
A
2
be two subspaces of
V
, and let
γ
1
,
γ
2
be
bases of
A
1
,
A
2
, respectively. Then
γ
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 Fall '07
 bieri
 Math, Linear Algebra, Algebra, Vector Space, Harvard University

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