University of Leipzig  SS19
10PHYBIPMA2  Mathematics 2 / Vitalii Konarovskyi
8.
[3 points]
Let a linear map
T
:
R
3
→
R
3
be defined by
T
(
x, y, z
) = (
x, x
+
y, x
+
y
+
z
). Find
its matrix in the standard basis and also in the basis
v
1
= (1
,
0
,
0),
v
2
= (2
,
0
,
1),
v
3
= (0
,
1
,
1).
Check if
T
is invertible and if yes, find the inverse map
T

1
. Is
T
selfadjoint?
9.
[3 points]
Find the orthogonal projection of the vector
u
= (4
,

1
,

3
,
4) onto span
{
v
1
, v
2
, v
3
}
in
R
4
with the standard inner product, where
v
1
= (1
,
1
,
1
,
1),
v
2
= (1
,
2
,
2
,

1),
v
3
= (1
,
0
,
0
,
3).
10.
[4 points]
Consider the vector space
R
n
[
x
] of all polynomials with real coefficients of degree at
most
n
with the inner product
h
p, q
i
=
Z
1

1
p
(
x
)
q
(
x
)
dx.
Show the following polynomials
p
0
(
x
) = 1
,
p
k
(
x
) =
1
2
k
k
!
d
k
dx
k
(
x
2

1)
k
,
k
= 1
, . . . , n,
form a basis in
R
n
[
x
].
11.
[3 points]
Show that
A
is selfadjoint and compute
e
A
, if
A
=
3

i
i
3
.
12.
[2 points]
Reduce to a canonical form the following quadratic form on
R
3
Q
(
x
) =
x
2
1

2
x
2
2
+
x
2
3
+ 2
x
1
x
2
+ 4
x
1
x
3
+ 2
x
2
x
3
.
13.
[2 points]
Show that
A
∪
B
=
A
∪
B
, where
A
denotes the closure of a set
A
.
14.
[3 points]
Compute the partial derivatives
∂f
∂x
(0
,
0) and
∂f
∂y
(0
,
0), if
f
(
x, y
) =
3
√
xy
.
Is the
function
f
differentiable at (0
,
0)?
15.
[3 points]
Does
∂
2
f
∂x∂y
(0
,
0) exist, if
f
(
x, y
) =
(
2
xy
x
2
+
y
2
,
if
x
2
+
y
2
>
0
,
0
,
if
x
=
y
= 0?
16.
[3 points]
Show that the function
z
=
z
(
x, y
) defined by the equation
F
(
x

az, y

bz
) = 0
,
where
F
is a differentiable function of two variables and
a, b
are some constants, solves the
equation
a
∂z
∂x
+
b
∂z
∂y
= 1
.
17.
[2+2+2 points]
Find the general solutions to the following equations:
a)
y
0
ln

y

+
x
2
y
= 0;
b)
xy
0
+ (1 + 2
x
2
)
y
=
x
3
e

x
2
;
c)
y
0
=
y
2
+2
xy
x
2
.
18.
[3+3 points]
Solve the initial value problems:
a)
y
0

xy
=
xy
3
2
,
y
(1) = 4;
b)
y
(4)

16
y
= 0,
y
(0) =
y
0
(0) = 2,
y
00
(0) =

2,
y
000
(0) = 0.
2