Exercise for the break Lay the
connecting arrow
from your left ear to the right small finger of the person in front of you, in a parallel way, at the tip of the nose of your left neighbor. What is the result?
Exercises Time is an
affine line
over
R
{\displaystyle {}\mathbb {R} }
connecting arrow
from the moment of your first milk tooth to the moment of your school enrollment at the moment now. What is the result?
Determine the parameter form for the line given by the equation
4
x
+
7
y
=
3
{\displaystyle {}4x+7y=3\,}
in
Q
2
{\displaystyle {}\mathbb {Q} ^{2}}
Let
V
{\displaystyle {}V}
vector space ,
U
⊆
V
{\displaystyle {}U\subseteq V}
linear subspace ,
and let
E
=
P
+
U
{\displaystyle {}E=P+U}
affine subspace .
Show that, for every point
Q
∈
E
{\displaystyle {}Q\in E}
E
=
Q
+
U
{\displaystyle {}E=Q+U}
Let
V
{\displaystyle {}V}
vector space ,
and let
E
⊆
V
{\displaystyle {}E\subseteq V}
affine subspace .
Show that
E
{\displaystyle {}E}
linear subspace
of
V
{\displaystyle {}V}
E
{\displaystyle {}E}
0
{\displaystyle {}0}
Let
φ
:
R
3
⟶
R
,
(
x
y
z
)
⟼
4
x
−
6
y
+
9
z
.
{\displaystyle \varphi \colon \mathbb {R} ^{3}\longrightarrow \mathbb {R} ,{\begin{pmatrix}x\\y\\z\end{pmatrix}}\longmapsto 4x-6y+9z.}
Determine, for the set
E
=
{
Q
∈
R
3
∣
φ
(
Q
)
=
5
}
,
{\displaystyle {}E={\left\{Q\in \mathbb {R} ^{3}\mid \varphi (Q)=5\right\}}\,,}
a description using a
starting point
and a
translation space.
Let
φ
:
R
3
⟶
R
2
,
(
x
y
z
)
⟼
(
7
x
+
y
−
3
z
4
x
+
5
y
)
.
{\displaystyle \varphi \colon \mathbb {R} ^{3}\longrightarrow \mathbb {R} ^{2},{\begin{pmatrix}x\\y\\z\end{pmatrix}}\longmapsto {\begin{pmatrix}7x+y-3z\\4x+5y\end{pmatrix}}.}
Determine, for the set
E
=
{
Q
∈
R
3
∣
φ
(
Q
)
=
(
4
−
2
)
}
,
{\displaystyle {}E={\left\{Q\in \mathbb {R} ^{3}\mid \varphi (Q)={\begin{pmatrix}4\\-2\end{pmatrix}}\right\}}\,,}
a description using a
starting point
and a
translating space.
We consider the three planes
E
,
F
,
G
{\displaystyle {}E,F,G}
Q
3
{\displaystyle {}\mathbb {Q} ^{3}}
E
=
{
(
x
,
y
,
z
)
∈
Q
3
∣
5
x
−
4
y
+
3
z
=
2
}
,
{\displaystyle {}E={\left\{(x,y,z)\in \mathbb {Q} ^{3}\mid 5x-4y+3z=2\right\}}\,,}
F
=
{
(
x
,
y
,
z
)
∈
Q
3
∣
7
x
−
5
y
+
6
z
=
3
}
,
{\displaystyle {}F={\left\{(x,y,z)\in \mathbb {Q} ^{3}\mid 7x-5y+6z=3\right\}}\,,}
G
=
{
(
x
,
y
,
z
)
∈
Q
3
∣
2
x
−
y
+
4
z
=
5
}
.
{\displaystyle {}G={\left\{(x,y,z)\in \mathbb {Q} ^{3}\mid 2x-y+4z=5\right\}}\,.}
Determine all points in
E
∩
F
∖
E
∩
F
∩
G
{\displaystyle {}E\cap F\setminus E\cap F\cap G}
Let
d
∈
N
+
{\displaystyle {}d\in \mathbb {N} _{+}}
K
{\displaystyle {}K}
field .
Let
n
{\displaystyle {}n}
a
1
,
…
,
a
n
∈
K
{\displaystyle {}a_{1},\ldots ,a_{n}\in K}
n
{\displaystyle {}n}
b
1
,
…
,
b
n
∈
K
{\displaystyle {}b_{1},\ldots ,b_{n}\in K}
E
{\displaystyle {}E}
P
{\displaystyle {}P}
d
{\displaystyle {}d}
P
(
a
i
)
=
b
i
{\displaystyle {}P(a_{i})=b_{i}\,}
for
i
=
1
,
…
,
n
{\displaystyle {}i=1,\ldots ,n}
affine subspace
of
K
[
X
]
≤
d
{\displaystyle {}K[X]_{\leq d}}
dimension
of
E
{\displaystyle {}E}
E
{\displaystyle {}E}
Let
E
{\displaystyle {}E}
affine space
over the
K
{\displaystyle {}K}
vector space
V
{\displaystyle {}V}
V
{\displaystyle {}V}
P
P
→
=
0
{\displaystyle {}{\overrightarrow {PP}}=0}
P
∈
E
{\displaystyle {}P\in E}
P
Q
→
=
−
Q
P
→
{\displaystyle {}{\overrightarrow {PQ}}=-{\overrightarrow {QP}}}
P
,
Q
∈
E
{\displaystyle {}P,Q\in E}
P
Q
→
+
Q
R
→
=
P
R
→
{\displaystyle {}{\overrightarrow {PQ}}+{\overrightarrow {QR}}={\overrightarrow {PR}}}
P
,
Q
,
R
∈
E
{\displaystyle {}P,Q,R\in E}
Show that the empty set is an
affine space
over any
K
{\displaystyle {}K}
vector space
V
{\displaystyle {}V}
Let
E
{\displaystyle {}E}
K
{\displaystyle {}K}
vector space
V
{\displaystyle {}V}
P
∈
E
{\displaystyle {}P\in E}
θ
:
V
⟶
E
,
v
⟼
P
+
v
,
{\displaystyle \theta \colon V\longrightarrow E,v\longmapsto P+v,}
be the corresponding bijection. Using this bijection, we identify
E
{\displaystyle {}E}
E
′
=
{
(
v
,
1
)
∈
V
×
K
∣
v
∈
V
}
{\displaystyle {}E'={\left\{(v,1)\in V\times K\mid v\in V\right\}}\,}
via the mapping
φ
:
E
⟶
E
′
,
P
⟼
(
θ
−
1
(
P
)
,
1
)
.
{\displaystyle \varphi \colon E\longrightarrow E',P\longmapsto (\theta ^{-1}(P),1).}
a) Show that
E
′
{\displaystyle {}E'}
affine subspace
of
V
×
K
{\displaystyle {}V\times K}
V
×
0
{\displaystyle {}V\times 0}
φ
(
Q
+
v
)
=
φ
(
Q
)
+
v
{\displaystyle {}\varphi (Q+v)=\varphi (Q)+v\,}
holds for all
Q
∈
E
{\displaystyle {}Q\in E}
Determine by a drawing the point that is given by the
barycentric combination
0
,
2
P
1
+
0
,
4
P
2
−
0
,
3
P
3
+
0
,
7
P
4
{\displaystyle 0,2P_{1}+0,4P_{2}-0,3P_{3}+0,7P_{4}}
in the image on the right. Start with different starting points.
Show that, for a family
P
i
{\displaystyle {}P_{i}}
i
∈
I
{\displaystyle {}i\in I}
affine space
E
{\displaystyle {}E}
barycentric combination
∑
i
∈
I
a
i
P
i
{\displaystyle \sum _{i\in I}a_{i}P_{i}}
defines a unique point in
E
{\displaystyle {}E}
Let
P
{\displaystyle {}P}
affine space
E
{\displaystyle {}E}
V
{\displaystyle {}V}
barycentric combinations
for
P
{\displaystyle {}P}
Q
∈
E
{\displaystyle {}Q\in E}
v
∈
V
{\displaystyle {}v\in V}
P
{\displaystyle {}P}
P
+
Q
−
Q
{\displaystyle {}P+Q-Q}
(
P
+
v
)
−
(
Q
+
v
)
+
Q
{\displaystyle {}(P+v)-(Q+v)+Q}
Q
P
→
{\displaystyle {}{\overrightarrow {QP}}}
P
−
Q
{\displaystyle {}P-Q}
Let
V
{\displaystyle {}V}
K
{\displaystyle {}K}
vector space ,
which we consider as an
affine space
over itself. Let
P
,
Q
∈
V
{\displaystyle {}P,Q\in V}
P
Q
→
=
Q
−
P
{\displaystyle {}{\overrightarrow {PQ}}=Q-P}
Let
E
{\displaystyle {}E}
affine space
over the
K
{\displaystyle {}K}
vector space
V
{\displaystyle {}V}
P
i
{\displaystyle {}P_{i}}
i
∈
I
{\displaystyle {}i\in I}
P
,
Q
{\displaystyle {}P,Q}
E
{\displaystyle {}E}
∑
i
∈
I
a
i
P
i
{\displaystyle {}\sum _{i\in I}a_{i}P_{i}}
barycentric combination .
Show that
P
−
Q
+
∑
i
∈
I
a
i
P
i
=
Q
P
→
+
(
∑
i
∈
I
a
i
P
i
)
{\displaystyle {}P-Q+\sum _{i\in I}a_{i}P_{i}={\overrightarrow {QP}}+{\left(\sum _{i\in I}a_{i}P_{i}\right)}\,}
holds, where the left-hand expression is a barycentric combination.
Let
V
{\displaystyle {}V}
vector space
over
K
{\displaystyle {}K}
affine space .
Let
∑
i
∈
I
a
i
v
i
{\displaystyle {}\sum _{i\in I}a_{i}v_{i}}
v
i
∈
V
{\displaystyle {}v_{i}\in V}
a
i
∈
K
{\displaystyle {}a_{i}\in K}
∑
i
∈
I
a
i
=
1
{\displaystyle {}\sum _{i\in I}a_{i}=1}
barycentric combination .
Show that the point defined by this barycentric combination in affine space equals the vector sum
∑
i
∈
I
a
i
v
i
{\displaystyle {}\sum _{i\in I}a_{i}v_{i}}
What are the
barycentric coordinates
of your favorite color in additive mixing?
Let
E
{\displaystyle {}E}
affine space
over a
K
{\displaystyle {}K}
vector space
V
{\displaystyle {}V}
P
1
,
…
,
P
n
{\displaystyle {}P_{1},\ldots ,P_{n}}
E
{\displaystyle {}E}
j
=
1
,
…
,
k
{\displaystyle {}j=1,\ldots ,k}
Q
j
=
∑
i
=
1
n
a
i
j
P
i
,
{\displaystyle {}Q_{j}=\sum _{i=1}^{n}a_{ij}P_{i}\,,}
with
∑
i
=
1
n
a
i
j
=
1
{\displaystyle {}\sum _{i=1}^{n}a_{ij}=1}
j
{\displaystyle {}j}
barycentric combinations
of the
P
i
{\displaystyle {}P_{i}}
b
1
,
…
,
b
k
∈
K
{\displaystyle {}b_{1},\ldots ,b_{k}\in K}
∑
j
=
1
k
b
j
=
1
{\displaystyle {}\sum _{j=1}^{k}b_{j}=1}
∑
j
=
1
k
b
j
Q
j
{\displaystyle \sum _{j=1}^{k}b_{j}Q_{j}}
as a barycentric combination of the
P
i
{\displaystyle {}P_{i}}
Imagine four points in the natural
3
{\displaystyle {}3}
affine basis
of the space.
Imagine four points in the natural
3
{\displaystyle {}3}
affine basis
of the space, but such that each three of these points form an affine basis in an affine plane.
Hand-in-exercises Let
φ
:
R
4
⟶
R
2
,
(
x
y
z
w
)
⟼
(
6
x
−
5
y
−
3
z
+
8
w
x
+
5
y
+
4
z
−
2
w
)
.
{\displaystyle \varphi \colon \mathbb {R} ^{4}\longrightarrow \mathbb {R} ^{2},{\begin{pmatrix}x\\y\\z\\w\end{pmatrix}}\longmapsto {\begin{pmatrix}6x-5y-3z+8w\\x+5y+4z-2w\end{pmatrix}}.}
Determine, for the set
E
=
{
Q
∈
R
4
∣
φ
(
Q
)
=
(
−
3
−
7
)
}
,
{\displaystyle {}E={\left\{Q\in \mathbb {R} ^{4}\mid \varphi (Q)={\begin{pmatrix}-3\\-7\end{pmatrix}}\right\}}\,,}
a description using a
starting point
and a
translating space.
Let
V
{\displaystyle {}V}
K
{\displaystyle {}K}
vector space .
We consider the set
E
=
{
(
v
,
1
)
∣
v
∈
V
}
⊂
V
×
K
,
{\displaystyle {}E={\left\{(v,1)\mid v\in V\right\}}\subset V\times K\,,}
which is an
affine space
over
V
{\displaystyle {}V}
a) Show that the points
P
i
=
(
v
i
,
1
)
{\displaystyle {}P_{i}=(v_{i},1)}
i
=
1
,
…
,
n
{\displaystyle {}i=1,\ldots ,n}
affine basis
of
E
{\displaystyle {}E}
P
i
{\displaystyle {}P_{i}}
V
×
K
{\displaystyle {}V\times K}
V
×
K
{\displaystyle {}V\times K}
P
∈
E
{\displaystyle {}P\in E}
barycentric coordinates
of
P
{\displaystyle {}P}
P
1
,
…
,
P
n
{\displaystyle {}P_{1},\ldots ,P_{n}}
P
{\displaystyle {}P}
P
1
,
…
,
P
n
{\displaystyle {}P_{1},\ldots ,P_{n}}
Let
E
{\displaystyle {}E}
F
{\displaystyle {}F}
affine spaces
over the
field
K
{\displaystyle {}K}
E
×
F
{\displaystyle {}E\times F}
Let
E
{\displaystyle {}E}
F
{\displaystyle {}F}
affine spaces
over the
field
K
{\displaystyle {}K}
affine basis
P
1
,
…
,
P
n
{\displaystyle {}P_{1},\ldots ,P_{n}}
E
{\displaystyle {}E}
Q
1
,
…
,
Q
m
{\displaystyle {}Q_{1},\ldots ,Q_{m}}
F
{\displaystyle {}F}
(
P
1
,
Q
1
)
,
(
P
1
,
Q
2
)
,
…
,
(
P
1
,
Q
m
)
,
(
P
2
,
Q
1
)
,
(
P
3
,
Q
1
)
,
…
,
(
P
n
,
Q
1
)
{\displaystyle (P_{1},Q_{1}),\,(P_{1},Q_{2}),\ldots ,(P_{1},Q_{m}),\,(P_{2},Q_{1}),\,(P_{3},Q_{1}),\ldots ,(P_{n},Q_{1})}
is an affine basis of the product space
E
×
F
{\displaystyle {}E\times F}
![{\displaystyle {}K[X]_{\leq d}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9632b5b5f5323aaba247da8c9275b2a50fe2dacd)
