Exercise for the break Determine the
dimension
of the space of all
m
×
n
{\displaystyle {}m\times n}
matrices .
Exercises Determine the
dimension
of the solution space of the
linear system
4
x
−
3
y
+
7
z
+
5
u
−
v
=
0
{\displaystyle {}4x-3y+7z+5u-v=0\,}
and
y
+
6
z
−
10
u
+
3
v
=
0
{\displaystyle {}y+6z-10u+3v=0\,}
in the variables
x
,
y
,
z
,
u
,
v
{\displaystyle {}x,y,z,u,v}
Let
K
{\displaystyle {}K}
field ,
and
n
∈
N
{\displaystyle {}n\in \mathbb {N} }
diagonal matrices
is a
linear subspace
in the space of all
n
×
n
{\displaystyle {}n\times n}
matrices
over
K
{\displaystyle {}K}
dimension .
An
n
×
n
{\displaystyle {}n\times n}
matrix
M
=
(
a
i
j
)
1
≤
i
,
j
≤
n
{\displaystyle {}M={\left(a_{ij}\right)}_{1\leq i,j\leq n}\,}
is called
symmetric
if
a
i
j
=
a
j
i
{\displaystyle {}a_{ij}=a_{ji}}
holds for all
i
,
j
{\displaystyle {}i,j}
.
Show that the set of all
symmetric
n
×
n
{\displaystyle {}n\times n}
matrices
is a
linear subspace
in the space of all
n
×
n
{\displaystyle {}n\times n}
dimension .
Let
K
{\displaystyle {}K}
field ,
and
n
∈
N
{\displaystyle {}n\in \mathbb {N} }
upper triangular matrices
is a
linear subspace
in the space of all
n
×
n
{\displaystyle {}n\times n}
matrices
over
K
{\displaystyle {}K}
dimension .
Let
K
{\displaystyle {}K}
V
{\displaystyle {}V}
K
{\displaystyle {}K}
n
=
dim
K
(
V
)
{\displaystyle {}n=\dim _{K}{\left(V\right)}}
n
{\displaystyle {}n}
v
1
,
…
,
v
n
{\displaystyle {}v_{1},\ldots ,v_{n}}
V
{\displaystyle {}V}
v
1
,
…
,
v
n
{\displaystyle {}v_{1},\ldots ,v_{n}}
V
{\displaystyle {}V}
v
1
,
…
,
v
n
{\displaystyle {}v_{1},\ldots ,v_{n}}
V
{\displaystyle {}V}
v
1
,
…
,
v
n
{\displaystyle {}v_{1},\ldots ,v_{n}}
Let
K
{\displaystyle {}K}
field ,
and let
V
{\displaystyle {}V}
K
{\displaystyle {}K}
vector space
of finite
dimension .
Let
U
⊆
V
{\displaystyle {}U\subseteq V}
linear subspace
with
dim
K
(
U
)
=
dim
K
(
V
)
{\displaystyle {}\dim _{K}{\left(U\right)}=\dim _{K}{\left(V\right)}}
U
=
V
{\displaystyle {}U=V}
Let
a
,
b
,
c
∈
R
{\displaystyle {}a,b,c\in \mathbb {R} }
(
a
b
c
)
,
(
c
a
b
)
,
(
b
c
a
)
∈
R
3
.
{\displaystyle {}{\begin{pmatrix}a\\b\\c\end{pmatrix}},\,{\begin{pmatrix}c\\a\\b\end{pmatrix}},\,{\begin{pmatrix}b\\c\\a\end{pmatrix}}\in \mathbb {R} ^{3}\,.}
Give examples for
a
,
b
,
c
{\displaystyle {}a,b,c}
0
,
1
,
2
,
3
{\displaystyle {}0,1,2,3}
Let
K
{\displaystyle {}K}
field ,
and let
V
{\displaystyle {}V}
W
{\displaystyle {}W}
finite-dimensional
K
{\displaystyle {}K}
vector spaces
with
dim
K
(
V
)
=
n
{\displaystyle {}\dim _{K}{\left(V\right)}=n\,}
and
dim
K
(
W
)
=
m
.
{\displaystyle {}\dim _{K}{\left(W\right)}=m\,.}
What is the
dimension
of the
Cartesian product
V
×
W
{\displaystyle {}V\times W}
Let
W
{\displaystyle {}W}
n
{\displaystyle {}n}
K
{\displaystyle {}K}
vector space
(
K
{\displaystyle {}K}
U
,
V
⊆
W
{\displaystyle {}U,V\subseteq W}
linear subspaces
of
dimension
dim
K
(
U
)
=
r
{\displaystyle {}\dim _{K}{\left(U\right)}=r}
dim
K
(
V
)
=
s
{\displaystyle {}\dim _{K}{\left(V\right)}=s}
r
+
s
>
n
{\displaystyle {}r+s>n}
U
∩
V
≠
0
{\displaystyle {}U\cap V\neq 0}
Let
K
{\displaystyle {}K}
K
[
X
]
{\displaystyle {}K[X]}
polynomial ring
over
K
{\displaystyle {}K}
d
∈
N
{\displaystyle {}d\in \mathbb {N} }
≤
d
{\displaystyle {}\leq d}
finite-dimensional
linear subspace
of
K
[
X
]
{\displaystyle {}K[X]}
dimension ?
Show that the set of all real
polynomials
of
degree
≤
4
{\displaystyle {}\leq 4}
−
2
{\displaystyle {}-2}
3
{\displaystyle {}3}
finite-dimensional
linear subspace
in
R
[
X
]
{\displaystyle {}\mathbb {R} [X]}
dimension .
Let
V
{\displaystyle {}V}
v
1
,
…
,
v
n
{\displaystyle {}v_{1},\ldots ,v_{n}}
basis
of
V
{\displaystyle {}V}
v
1
,
…
,
v
n
and
i
v
1
,
…
,
i
v
n
{\displaystyle v_{1},\ldots ,v_{n}{\text{ and }}{\mathrm {i} }v_{1},\ldots ,{\mathrm {i} }v_{n}}
forms a basis for
V
{\displaystyle {}V}
Consider the standard basis
e
1
,
e
2
,
e
3
,
e
4
{\displaystyle {}e_{1},e_{2},e_{3},e_{4}}
R
4
{\displaystyle {}\mathbb {R} ^{4}}
(
1
3
0
−
4
)
,
(
2
1
5
7
)
and
(
−
4
9
−
5
1
)
.
{\displaystyle {\begin{pmatrix}1\\3\\0\\-4\end{pmatrix}},\,{\begin{pmatrix}2\\1\\5\\7\end{pmatrix}}{\text{ and }}{\begin{pmatrix}-4\\9\\-5\\1\end{pmatrix}}.}
Prove that these vectors are linearly independent, and extend them to a basis by adding an appropriate standard vector, as shown in the
base exchange theorem .
Can one take any standard vector?
We consider the
linear equations
9
x
−
8
y
+
7
z
−
8
u
+
4
v
=
0
,
{\displaystyle {}9x-8y+7z-8u+4v=0\,,}
3
y
+
7
z
−
4
u
+
6
v
=
0
,
{\displaystyle {}\,\,\,\,\,\,\,\,\,\,\,\,\,\,3y+7z-4u+6v=0\,,}
−
2
z
+
5
u
+
7
v
=
0
,
{\displaystyle {}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,-2z+5u+7v=0\,,}
over
R
{\displaystyle {}\mathbb {R} }
Determine a
basis
b
1
{\displaystyle {}{\mathfrak {b}}_{1}}
Extend the basis
b
1
{\displaystyle {}{\mathfrak {b}}_{1}}
b
2
{\displaystyle {}{\mathfrak {b}}_{2}}
Extend the basis
b
2
{\displaystyle {}{\mathfrak {b}}_{2}}
b
3
{\displaystyle {}{\mathfrak {b}}_{3}}
Extend the basis
b
3
{\displaystyle {}{\mathfrak {b}}_{3}}
b
4
{\displaystyle {}{\mathfrak {b}}_{4}}
R
5
{\displaystyle {}\mathbb {R} ^{5}}
We consider the last digit in the basic multiplication table as a family of
9
{\displaystyle {}9}
9
{\displaystyle {}9}
(
1
2
3
4
5
6
7
8
9
2
4
6
8
0
2
4
6
8
3
6
9
2
5
8
1
4
7
4
8
2
6
0
4
8
2
6
5
0
5
0
5
0
5
0
5
6
2
8
4
0
6
2
8
4
7
4
1
8
5
2
9
6
3
8
6
4
2
0
8
6
4
2
9
8
7
6
5
4
3
2
1
)
.
{\displaystyle {\begin{pmatrix}1&2&3&4&5&6&7&8&9\\2&4&6&8&0&2&4&6&8\\3&6&9&2&5&8&1&4&7\\4&8&2&6&0&4&8&2&6\\5&0&5&0&5&0&5&0&5\\6&2&8&4&0&6&2&8&4\\7&4&1&8&5&2&9&6&3\\8&6&4&2&0&8&6&4&2\\9&8&7&6&5&4&3&2&1\end{pmatrix}}.}
What is the
dimension
of the
linear subspace
in
R
9
{\displaystyle {}\mathbb {R} ^{9}}
Let
K
{\displaystyle {}K}
field ,
and let
V
{\displaystyle {}V}
K
{\displaystyle {}K}
vector space .
Show that
V
{\displaystyle {}V}
basis
and an infinite basis.
The magic square in Dürer's picture Melencolia I.
In this sense, the matrix
(
c
0
⋯
0
0
c
⋱
⋮
⋮
⋱
⋱
0
0
⋯
0
c
)
{\displaystyle {\begin{pmatrix}c&0&\cdots &0\\0&c&\ddots &\vdots \\\vdots &\ddots &\ddots &0\\0&\cdots &0&c\end{pmatrix}}}
is, for every
c
∈
K
{\displaystyle {}c\in K}
Show that the set of all
linear-magic squares
of length
n
{\displaystyle {}n}
field
K
{\displaystyle {}K}
linear subspace
in the space of all
n
×
n
{\displaystyle {}n\times n}
matrices .
Hand-in-exercises Let
K
{\displaystyle {}K}
V
{\displaystyle {}V}
K
{\displaystyle {}K}
v
1
,
…
,
v
m
{\displaystyle {}v_{1},\ldots ,v_{m}}
V
{\displaystyle {}V}
U
=
⟨
v
i
,
i
=
1
,
…
,
m
⟩
{\displaystyle {}U=\langle v_{i},\,i=1,\ldots ,m\rangle \,}
be the
linear subspace
they span. Prove that the family is linearly independent if and only if the dimension of
U
{\displaystyle {}U}
m
{\displaystyle {}m}
a) Determine the
dimension
of the solution space of the
linear system
2
x
+
5
y
+
7
z
+
4
u
−
3
v
+
2
w
=
0
{\displaystyle {}2x+5y+7z+4u-3v+2w=0\,}
4
x
+
9
y
+
6
z
+
5
u
−
v
+
w
=
0
{\displaystyle {}4x+9y+6z+5u-v+w=0\,}
7
x
+
8
y
−
3
z
+
u
+
3
v
+
3
w
=
0
{\displaystyle {}7x+8y-3z+u+3v+3w=0\,}
−
x
+
6
y
+
16
z
+
8
u
−
7
v
=
0
{\displaystyle {}-x+6y+16z+8u-7v=0\,}
in the variables
x
,
y
,
z
,
u
,
v
,
w
{\displaystyle {}x,y,z,u,v,w}
x
,
y
,
z
,
u
,
v
,
w
,
r
,
s
{\displaystyle {}x,y,z,u,v,w,r,s}
Show that the set of all real
polynomials
of
degree
≤
6
{\displaystyle {}\leq 6}
−
1
{\displaystyle {}-1}
0
{\displaystyle {}0}
1
{\displaystyle {}1}
finite-dimensional
subspace
of
R
[
X
]
{\displaystyle {}\mathbb {R} [X]}
dimension
of this vector space.
Let
K
{\displaystyle {}K}
field ,
and
n
∈
N
+
{\displaystyle {}n\in \mathbb {N} _{+}}
dimension
of the space of all
linear-magic squares
of length
n
{\displaystyle {}n}
K
{\displaystyle {}K}
We consider the
linear equations
8
x
−
3
y
+
5
z
+
7
u
+
6
v
=
0
,
{\displaystyle {}8x-3y+5z+7u+6v=0\,,}
9
x
+
2
y
+
z
−
v
=
0
,
{\displaystyle {}9x+2y+\,\,z\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,-v=0\,,}
7
y
−
z
+
4
u
=
0
,
{\displaystyle {}\,\,\,\,\,\,\,\,\,\,\,\,\,\,7y-\,\,z+4u\,\,\,\,\,\,\,\,\,\,\,\,\,\,=0\,,}
over
R
{\displaystyle {}\mathbb {R} }
Determine a
basis
b
1
{\displaystyle {}{\mathfrak {b}}_{1}}
Extend the basis
b
1
{\displaystyle {}{\mathfrak {b}}_{1}}
b
2
{\displaystyle {}{\mathfrak {b}}_{2}}
Extend the basis
b
2
{\displaystyle {}{\mathfrak {b}}_{2}}
b
3
{\displaystyle {}{\mathfrak {b}}_{3}}
Extend the basis
b
3
{\displaystyle {}{\mathfrak {b}}_{3}}
b
4
{\displaystyle {}{\mathfrak {b}}_{4}}
R
5
{\displaystyle {}\mathbb {R} ^{5}}
![{\displaystyle {}K[X]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7861df9df3d7463835d927c4d0d4c27f5e2f8416)
![{\displaystyle {}\mathbb {R} [X]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/52a170b822ae65cbe9d1cb122f493fcd88305b7f)
.jpg)
