AMS210.01.
Homework 7
Andrei Antonenko
This homework is optional. You can solve it to get some extrapoints in this class (contact me to
get them). I recommend you to solve problems from it — this will really help you during your final!
1. Let the coordinates of the vector
v
in the “old” basis are
v
= (6
,
9
,
14). Find the coordinates of
the vector in the “new” basis
{
e
1
= (1
,
1
,
1)
, e
2
= (1
,
1
,
2)
, e
3
= (1
,
2
,
3)
}
.
2. Let the “old” basis is
{
e
1
= (1
,
2
,
1)
, e
2
= (2
,
3
,
3)
, e
3
= (3
,
8
,
9)
}
, and “new” basis is
{
e
0
1
=
(3
,
5
,
8)
, e
0
2
= (5
,
14
,
13)
, e
0
3
= (1
,
9
,
2)
}
. Find the changeofbasis matrix.
3. How does the changeofbasis matrix change if we
(a) interchange two vectors of the “old” basis?
(b) interchange two vectors of the “new” basis?
(c) take vectors of both bases in reverse order?
4. Determine, which of the following mappings are linear operators. Justify your answer.
(a)
x
7→
a
, where
a
is a fixed vector.
(b)
x
7→
x
+
a
, where
a
is a fixed vector.
(c)
x
7→
αx
, where
α
is a given number.
(d)
x
7→ h
x, a
i
b
, where
a, b
are fixed vectors, and
V
is a Euclidean space.
(e)
f
(
x
)
7→
f
(
x
+ 2), where
f
is a function.
(f)
f
(
x
)
7→
f
(
x
+ 1)

f
(
x
), where
f
is a function.
(g) (
x
1
, x
2
, x
3
)
7→
(
x
1
+ 2
, x
2
+ 5
, x
3
).
(h) (
x
1
, x
2
, x
3
)
7→
(
x
1
, x
2
, x
1
+
x
2
+
x
3
).
5. Find the matrix of the following linear operators:
(a) (
x
1
, x
2
, x
3
)
7→
(
x
1
, x
1
+ 2
x
2
, x
2
+ 3
x
3
) in the standard basis.
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 Spring '03
 Andant
 Linear Algebra, basis, Andrei Antonenko

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