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Linear Algebra Quiz 3
June 13, 2006
1.
Consider
R
2
with ordered bases
β
=
{
2
1
,
1
1
}
and
γ
=
{
3
1
,
2
1
}
.
(a)(4 points) Find the transition matrices from
β
-coordinates to standard coordinates,
γ
-
coordinates to standard coordinates, and
γ
-coordinates to
β
-coordinates, i.e., find [id]
STD
β
,
[id]
STD
γ
, and [id]
γ
β
.
(b) (4 points) Let
L
:
R
2
→
R
2
by
L
x
y
=
2
x
+
y
-
x
+ 3
y
. Show that
L
is a linear transfor-
mation and find a matrix
A
such that
L
(
x
) =
Ax
for all
x
∈
R
2
.
(c) (4 points) Find the matrix representing
L
with respect to the bases
β
to
β
and
β
to
γ
,
i.e., find [
L
]
β
β
and [
L
]
γ
β
.
(d) (4 points) Suppose
x
= 2
3
1
-
2
1
.
Find the coordinate representation of
x
with
respect to the ordered bases
β
,
γ
, and the standard basis, i.e., find [
x
]
β
, [
x
]
γ
, and [
x
]
STD
.
(e) (4 points) Find the coordinate representation of
L
(
x
) with respect to the ordered bases
β
,
γ
, and the standard basis, i.e., find [
L
(
x
)]
β
, [
L
(
x
)]
γ
, and [
L
(
x
)]
STD
.
Solution:
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