Sample questions for Prelim 2
Math 2940
Fall 2009
This represents relevant questions that have appeared on previous prelims
and finals. The overall length is not representative of a single prelim.
1. Let
T
:
R
3
→
P
4
be defined by
T
a
b
c
=
p
(
x
) = (
a

2
b
+3
c
)+(3
a
+2
b
+
c
)
x
+(
a
+2
b

c
)
x
2
+(
a
+
c
)
x
4
(a) Find the dimension and a basis for im(
T
).
(b) Find the coordinates of the polynomial
p
(
x
) above in terms of
your basis for im(
T
).
(c) Find the dimension and a basis for ker(
T
).
2. In each of the following, you are given a linear space
V
and a subset
W
⊆
V
. Decide whether
W
is a subspace of
V
, and prove your answer
is correct.
(a)
V
is the space
R
2
×
2
of all 2
×
2 matrices, and
W
is the set of 2
×
2
matrices
A
such that
A
2
=
A
.
(b)
V
is the space of differentiable functions, and
W
is the set of those
differentiable functions that satisfy
f
(3) = 0.
3.
(a) Find a quadratic function of the form
f
(
x
) =
c
+
dx
2
that best fits
the data (
x, y
) = (

1
,
1)
,
(0
,
1)
,
(1
,
2) in the least squares sense.
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 Spring '06
 PANTANO
 Math, Linear Algebra, Multivariable Calculus, constant function, better fit, squares sense

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