AMS210.01.
Homework 4
Andrei Antonenko
Due at the beginning of the class, March 28, 2003
1. Let
f
:
R
3
→
R
2
such that
f
(
x, y, z
) = (
x
+ 2
z,
2
x

y

z
)
and
g
:
R
2
→
R
3
such that
g
(
x, y
) = (2
x
+
y, x
+ 2
y, x
+
y
)
.
Find (
f
◦
g
) and (
g
◦
f
).
2. Check which of the following functions are linear. Justify your answer.
(a)
f
(
x, y
) = (
x, y
2
)
(b)
f
(
x, y
) = (

x
+
y

, x

y
)
(c)
f
(
x, y, z
) = (
x
+
y, z,
0)
(d)
f
(
x, y
) = ((
x
+ 1)
2
, y
+ 3)
(e)
f
(
x, y, z
) = (
x
+
y
+
z, y
+
z, z
)
3. Let
f
:
M
n,n
→
R
be a function which maps any
n
×
n
matrix to a sum of its diagonal elements.
Determine is this function linear or not.
4. Let
V
be a vector space of all
n
×
n
matrices, and
M
be a fixed matrix in
V
.
Which of the
following functions
T
i
:
V
→
V
are linear:
(a)
T
1
(
X
) =
XM
(b)
T
2
(
X
) =
X
+
M
(c)
T
3
(
X
) =
XM

MX
(d) If
M
is invertible,
T
4
(
X
) =
MXM

1
5. Let
f
:
R
3
→
R
2
such that
f
(
x, y, z
) = (2
x
+
y, x
+
y
+
z
).
(a) Check that
f
is a linear function.
(b) Find the dimension of the kernel of
f
and its basis.
(c) Find the dimension of the image of
f
and its basis.
1
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(d) Find the matrix of this function in standard basis.
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 Spring '03
 Andant
 Linear Algebra, Derivative, Vector Space, Linear map

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