Exercises Compute the following product of matrices
(
Z
E
I
L
E
R
E
I
H
E
H
O
R
I
Z
O
N
T
A
L
)
⋅
(
S
E
I
P
V
K
A
E
A
L
R
A
T
T
L
)
.
{\displaystyle {\begin{pmatrix}Z&E&I&L&E\\R&E&I&H&E\\H&O&R&I&Z\\O&N&T&A&L\end{pmatrix}}\cdot {\begin{pmatrix}S&E&I\\P&V&K\\A&E&A\\L&R&A\\T&T&L\end{pmatrix}}.}
Compute, over the complex numbers, the following product of matrices
(
2
−
i
−
1
−
3
i
−
1
i
0
4
−
2
i
)
(
1
+
i
1
−
i
2
+
5
i
)
.
{\displaystyle {\begin{pmatrix}2-{\mathrm {i} }&-1-3{\mathrm {i} }&-1\\{\mathrm {i} }&0&4-2{\mathrm {i} }\end{pmatrix}}{\begin{pmatrix}1+{\mathrm {i} }\\1-{\mathrm {i} }\\2+5{\mathrm {i} }\end{pmatrix}}.}
Determine the product of matrices
e
i
∘
e
j
,
{\displaystyle e_{i}\circ e_{j},}
where the
i
{\displaystyle {}i}
n
{\displaystyle {}n}
j
{\displaystyle {}j}
n
{\displaystyle {}n}
Let
M
{\displaystyle {}M}
m
×
n
{\displaystyle {}m\times n}
M
e
j
{\displaystyle {}Me_{j}}
M
{\displaystyle {}M}
j
{\displaystyle {}j}
j
{\displaystyle {}j}
M
{\displaystyle {}M}
e
i
M
{\displaystyle {}e_{i}M}
e
i
{\displaystyle {}e_{i}}
i
{\displaystyle {}i}
Let
D
=
(
d
11
0
⋯
⋯
0
0
d
22
0
⋯
0
⋮
⋱
⋱
⋱
⋮
0
⋯
0
d
n
−
1
n
−
1
0
0
⋯
⋯
0
d
n
n
)
{\displaystyle {}D={\begin{pmatrix}d_{11}&0&\cdots &\cdots &0\\0&d_{22}&0&\cdots &0\\\vdots &\ddots &\ddots &\ddots &\vdots \\0&\cdots &0&d_{n-1\,n-1}&0\\0&\cdots &\cdots &0&d_{nn}\end{pmatrix}}\,}
be a
diagonal matrix ,
and
M
{\displaystyle {}M}
n
×
n
{\displaystyle {}n\times n}
D
M
{\displaystyle {}DM}
M
D
{\displaystyle {}MD}
Let
D
=
(
d
11
0
⋯
⋯
0
0
d
22
0
⋯
0
⋮
⋱
⋱
⋱
⋮
0
⋯
0
d
n
−
1
n
−
1
0
0
⋯
⋯
0
d
n
n
)
{\displaystyle {}D={\begin{pmatrix}d_{11}&0&\cdots &\cdots &0\\0&d_{22}&0&\cdots &0\\\vdots &\ddots &\ddots &\ddots &\vdots \\0&\cdots &0&d_{n-1\,n-1}&0\\0&\cdots &\cdots &0&d_{nn}\end{pmatrix}}\,}
be a
diagonal matrix ,
and let
c
=
(
c
1
⋮
c
n
)
{\displaystyle {}c={\begin{pmatrix}c_{1}\\\vdots \\c_{n}\end{pmatrix}}}
n
{\displaystyle {}n}
field
K
{\displaystyle {}K}
x
=
(
x
1
⋮
x
n
)
{\displaystyle {}x={\begin{pmatrix}x_{1}\\\vdots \\x_{n}\end{pmatrix}}}
D
x
=
c
,
{\displaystyle {}Dx=c\,,}
and how can you solve it?
Compute the product of matrices
(
2
+
i
1
−
1
2
i
4
i
−
5
+
7
i
2
+
i
0
)
(
−
5
+
4
i
3
−
2
i
2
−
i
e
+
π
i
1
−
i
)
(
1
+
i
2
−
3
i
)
,
{\displaystyle {\begin{pmatrix}2+{\mathrm {i} }&1-{\frac {1}{2}}{\mathrm {i} }&4{\mathrm {i} }\\-5+7{\mathrm {i} }&{\sqrt {2}}+{\mathrm {i} }&0\end{pmatrix}}{\begin{pmatrix}-5+4{\mathrm {i} }&3-2{\mathrm {i} }\\{\sqrt {2}}-{\mathrm {i} }&e+\pi {\mathrm {i} }\\1&-{\mathrm {i} }\end{pmatrix}}{\begin{pmatrix}1+{\mathrm {i} }\\2-3{\mathrm {i} }\end{pmatrix}},}
according to the two possible parentheses.
Show that the multiplication of matrices is associative. More precisely: Let
K
{\displaystyle {}K}
field ,
and let
A
{\displaystyle {}A}
m
×
n
{\displaystyle {}m\times n}
B
{\displaystyle {}B}
n
×
p
{\displaystyle {}n\times p}
C
{\displaystyle {}C}
p
×
r
{\displaystyle {}p\times r}
K
{\displaystyle {}K}
(
A
B
)
C
=
A
(
B
C
)
{\displaystyle {}(AB)C=A(BC)}
Show that the
matrix multiplication
of square matrices is, in general, not
commutative .
M
{\displaystyle {}M}
M
n
{\displaystyle {}M^{n}}
n
{\displaystyle {}n}
M
{\displaystyle {}M}
M
n
{\displaystyle {}M^{n}}
n
{\displaystyle {}n}
power of the matrix.
Compute, for the matrix
M
=
(
2
4
6
1
3
5
0
1
2
)
,
{\displaystyle {}M={\begin{pmatrix}2&4&6\\1&3&5\\0&1&2\end{pmatrix}}\,,}
the powers
M
i
,
i
=
1
,
2
,
3
,
4.
{\displaystyle M^{i},\,i=1,2,3,4.}
Out of the resources
R
1
,
R
2
{\displaystyle {}R_{1},R_{2}}
R
3
{\displaystyle {}R_{3}}
P
1
,
P
2
,
P
3
,
P
4
{\displaystyle {}P_{1},P_{2},P_{3},P_{4}}
R
1
{\displaystyle {}R_{1}}
R
2
{\displaystyle {}R_{2}}
R
3
{\displaystyle {}R_{3}}
P
1
{\displaystyle {}P_{1}}
6
{\displaystyle {}6}
2
{\displaystyle {}2}
3
{\displaystyle {}3}
P
2
{\displaystyle {}P_{2}}
4
{\displaystyle {}4}
1
{\displaystyle {}1}
2
{\displaystyle {}2}
P
3
{\displaystyle {}P_{3}}
0
{\displaystyle {}0}
5
{\displaystyle {}5}
2
{\displaystyle {}2}
P
4
{\displaystyle {}P_{4}}
2
{\displaystyle {}2}
1
{\displaystyle {}1}
5
{\displaystyle {}5}
a) Establish a matrix that computes, applied to a four-tuple of commodities, the required resources.
P
1
{\displaystyle {}P_{1}}
P
2
{\displaystyle {}P_{2}}
P
3
{\displaystyle {}P_{3}}
P
4
{\displaystyle {}P_{4}}
6
{\displaystyle {}6}
4
{\displaystyle {}4}
7
{\displaystyle {}7}
5
{\displaystyle {}5}
What resources are necessary?
R
1
{\displaystyle {}R_{1}}
R
2
{\displaystyle {}R_{2}}
R
3
{\displaystyle {}R_{3}}
12
{\displaystyle {}12}
9
{\displaystyle {}9}
13
{\displaystyle {}13}
What tuples of commodities can be produced from this without waste?
Determine
(approximately)
the coordinates of the sketched point
(the side length of a box represents a unit).
Draw the following points in the Cartesian plane
R
2
{\displaystyle {}\mathbb {R} ^{2}}
(
3
,
−
7
)
,
(
−
1
,
−
2
)
,
(
0
,
5
)
,
(
4
,
4
)
,
(
4
,
5
)
,
(
−
3
,
0
)
,
(
0
,
0
)
.
{\displaystyle (3,-7),\,(-1,-2),\,(0,5),\,(4,4),\,(4,5),\,(-3,0),\,(0,0).}
Let a point
P
=
(
x
,
y
)
{\displaystyle {}P=(x,y)}
R
2
{\displaystyle {}\mathbb {R} ^{2}}
(
−
x
,
y
)
,
(
x
,
−
y
)
,
(
−
x
,
−
y
)
,
(
3
x
,
3
y
)
,
(
−
2
x
,
−
2
y
)
.
{\displaystyle (-x,y),\,(x,-y),\,(-x,-y),\,(3x,3y),(-2x,-2y).}
Let a point
P
=
(
x
,
y
)
{\displaystyle {}P=(x,y)}
R
2
{\displaystyle {}\mathbb {R} ^{2}}
(
c
x
,
c
y
)
,
c
∈
R
.
{\displaystyle (cx,cy),\,c\in \mathbb {R} .}
Draw two points
P
{\displaystyle {}P}
Q
{\displaystyle {}Q}
R
2
{\displaystyle {}\mathbb {R} ^{2}}
Show that the product space
K
n
{\displaystyle {}K^{n}}
field
K
{\displaystyle {}K}
r
(
s
u
)
=
(
r
s
)
u
{\displaystyle {}r(su)=(rs)u}
r
(
u
+
v
)
=
r
u
+
r
v
{\displaystyle {}r(u+v)=ru+rv}
(
r
+
s
)
u
=
r
u
+
s
u
{\displaystyle {}(r+s)u=ru+su}
1
u
=
u
,
{\displaystyle {}1u=u\,,}
hold.
Let
K
{\displaystyle {}K}
field ,
and let
V
{\displaystyle {}V}
W
{\displaystyle {}W}
vector spaces
over
K
{\displaystyle {}K}
product set
V
×
W
{\displaystyle V\times W}
is also a
K
{\displaystyle {}K}
Let
V
{\displaystyle {}V}
vector space
over a
field
K
{\displaystyle {}K}
s
1
,
…
,
s
k
∈
K
{\displaystyle {}s_{1},\ldots ,s_{k}\in K}
v
1
,
…
,
v
n
∈
V
{\displaystyle {}v_{1},\ldots ,v_{n}\in V}
(
∑
i
=
1
k
s
i
)
⋅
(
∑
j
=
1
n
v
j
)
=
∑
1
≤
i
≤
k
,
1
≤
j
≤
n
s
i
⋅
v
j
.
{\displaystyle {}{\left(\sum _{i=1}^{k}s_{i}\right)}\cdot {\left(\sum _{j=1}^{n}v_{j}\right)}=\sum _{1\leq i\leq k,\,1\leq j\leq n}s_{i}\cdot v_{j}\,.}
Show that the addition and the scalar multiplication of a
vector space
V
{\displaystyle {}V}
linear subspace ,
and that this subspace with the inherited structures of
V
{\displaystyle {}V}
Check whether the following subsets of
R
2
{\displaystyle {}\mathbb {R} ^{2}}
linear subspaces :
V
1
=
{
(
x
,
y
)
∈
R
2
∣
x
+
2
y
=
0
}
{\displaystyle {}V_{1}={\left\{(x,y)\in \mathbb {R} ^{2}\mid x+2y=0\right\}}}
V
2
=
{
(
x
,
y
)
∈
R
2
∣
x
≥
y
}
{\displaystyle {}V_{2}={\left\{(x,y)\in \mathbb {R} ^{2}\mid x\geq y\right\}}}
V
3
=
{
(
x
,
y
)
∈
R
2
∣
y
=
x
+
1
}
{\displaystyle {}V_{3}={\left\{(x,y)\in \mathbb {R} ^{2}\mid y=x+1\right\}}}
V
4
=
{
(
x
,
y
)
∈
R
2
∣
x
y
=
0
}
{\displaystyle {}V_{4}={\left\{(x,y)\in \mathbb {R} ^{2}\mid xy=0\right\}}}
Let
K
{\displaystyle {}K}
field ,
and let
a
11
x
1
+
a
12
x
2
+
⋯
+
a
1
n
x
n
=
0
a
21
x
1
+
a
22
x
2
+
⋯
+
a
2
n
x
n
=
0
⋮
⋮
⋮
a
m
1
x
1
+
a
m
2
x
2
+
⋯
+
a
m
n
x
n
=
0
{\displaystyle {\begin{matrix}a_{11}x_{1}+a_{12}x_{2}+\cdots +a_{1n}x_{n}&=&0\\a_{21}x_{1}+a_{22}x_{2}+\cdots +a_{2n}x_{n}&=&0\\\vdots &\vdots &\vdots \\a_{m1}x_{1}+a_{m2}x_{2}+\cdots +a_{mn}x_{n}&=&0\end{matrix}}}
be a system of linear equations over
K
{\displaystyle {}K}
linear subspace
of
K
n
{\displaystyle {}K^{n}}
Let
D
{\displaystyle {}D}
2
×
2
{\displaystyle {}2\times 2}
(
a
11
a
12
a
21
a
22
)
,
{\displaystyle {\begin{pmatrix}a_{11}&a_{12}\\a_{21}&a_{22}\end{pmatrix}},}
which fulfill the condition
a
11
a
22
−
a
21
a
12
=
0
.
{\displaystyle {}a_{11}a_{22}-a_{21}a_{12}=0\,.}
Show that
D
{\displaystyle {}D}
linear subspace
in the space of all
2
×
2
{\displaystyle {}2\times 2}
Let
K
{\displaystyle {}K}
field ,
and let
V
{\displaystyle {}V}
K
{\displaystyle {}K}
vector space .
Let
U
,
W
⊆
V
{\displaystyle {}U,W\subseteq V}
linear subspaces
of
V
{\displaystyle {}V}
U
∪
W
{\displaystyle {}U\cup W}
V
{\displaystyle {}V}
U
⊆
W
{\displaystyle {}U\subseteq W}
W
⊆
U
{\displaystyle {}W\subseteq U}
Let
K
{\displaystyle {}K}
field ,
and
I
{\displaystyle {}I}
K
I
:=
Maps
(
I
,
K
)
,
{\displaystyle {}K^{I}:=\operatorname {Maps} \,(I,K)\,,}
with pointwise addition and scalar multiplication, is a
K
{\displaystyle {}K}
vector space .
Let
C
=
{
(
x
n
)
n
∈
N
∣
Cauchy sequence in
R
}
{\displaystyle {}C={\left\{{\left(x_{n}\right)}_{n\in \mathbb {N} }\mid {\text{Cauchy sequence in }}\mathbb {R} \right\}}\,}
be the set of all
real Cauchy sequences.
Show that
C
{\displaystyle {}C}
linear subspace
of the space of all sequences
F
=
{
(
x
n
)
n
∈
N
∣
sequence in
R
}
.
{\displaystyle {}F={\left\{{\left(x_{n}\right)}_{n\in \mathbb {N} }\mid {\text{sequence in }}\mathbb {R} \right\}}\,.}
Show that the subset
S
=
{
f
:
R
→
R
∣
f
continuous
}
⊆
Map
(
R
,
R
)
{\displaystyle {}S={\left\{f:\mathbb {R} \rightarrow \mathbb {R} \mid f{\text{ continuous}}\right\}}\subseteq \operatorname {Map} \,{\left(\mathbb {R} ,\mathbb {R} \right)}\,}
is a
linear subspace .
Show that the subset
S
=
{
f
:
R
→
R
∣
f
differentiable
}
⊆
Map
(
R
,
R
)
{\displaystyle {}S={\left\{f:\mathbb {R} \rightarrow \mathbb {R} \mid f{\text{ differentiable}}\right\}}\subseteq \operatorname {Map} \,{\left(\mathbb {R} ,\mathbb {R} \right)}\,}
is a
linear subspace .
Show that the subset
M
=
{
f
:
R
→
R
∣
f
monotonic
}
⊆
Map
(
R
,
R
)
{\displaystyle {}M={\left\{f:\mathbb {R} \rightarrow \mathbb {R} \mid f{\text{ monotonic}}\right\}}\subseteq \operatorname {Map} \,{\left(\mathbb {R} ,\mathbb {R} \right)}\,}
is not a
linear subspace .
Hand-in-exercises Compute, over the complex numbers, the following product of matrices
(
3
−
2
i
1
+
5
i
0
7
i
2
+
i
4
−
i
)
(
1
−
2
i
−
i
3
−
4
i
2
+
3
i
5
−
7
i
2
−
i
)
.
{\displaystyle {\begin{pmatrix}3-2{\mathrm {i} }&1+5{\mathrm {i} }&0\\7{\mathrm {i} }&2+{\mathrm {i} }&4-{\mathrm {i} }\end{pmatrix}}{\begin{pmatrix}1-2{\mathrm {i} }&-{\mathrm {i} }\\3-4{\mathrm {i} }&2+3{\mathrm {i} }\\5-7{\mathrm {i} }&2-{\mathrm {i} }\end{pmatrix}}.}
We consider the matrix
M
=
(
0
a
b
c
0
0
d
e
0
0
0
f
0
0
0
0
)
{\displaystyle {}M={\begin{pmatrix}0&a&b&c\\0&0&d&e\\0&0&0&f\\0&0&0&0\end{pmatrix}}\,}
over a field
K
{\displaystyle {}K}
M
{\displaystyle {}M}
0
{\displaystyle {}0}
M
4
=
M
M
M
M
=
0
.
{\displaystyle {}M^{4}=MMMM=0\,.}
Let
n
∈
N
{\displaystyle {}n\in \mathbb {N} }
n
{\displaystyle {}n}
(
a
b
0
c
)
.
{\displaystyle {\begin{pmatrix}a&b\\0&c\end{pmatrix}}.}
Find, appart from the matrices
(
1
0
0
1
)
{\displaystyle {}{\begin{pmatrix}1&0\\0&1\end{pmatrix}}}
(
−
1
0
0
−
1
)
{\displaystyle {}{\begin{pmatrix}-1&0\\0&-1\end{pmatrix}}}
M
{\displaystyle {}M}
M
2
=
(
1
0
0
1
)
{\displaystyle {}M^{2}={\begin{pmatrix}1&0\\0&1\end{pmatrix}}}
Let
K
{\displaystyle {}K}
field ,
and let
V
{\displaystyle {}V}
K
{\displaystyle {}K}
vector space .
Show that the following properties hold
(for
v
∈
V
{\displaystyle {}v\in V}
s
∈
K
{\displaystyle {}s\in K}
We have
0
v
=
0
{\displaystyle {}0v=0}
We have
s
0
=
0
{\displaystyle {}s0=0}
We have
(
−
1
)
v
=
−
v
{\displaystyle {}(-1)v=-v}
If
s
≠
0
{\displaystyle {}s\neq 0}
v
≠
0
{\displaystyle {}v\neq 0}
s
v
≠
0
{\displaystyle {}sv\neq 0}
Give an example of a vector space
V
{\displaystyle {}V}
V
{\displaystyle {}V}
