Exercise for the break Lay the
connecting arrow MDLD/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 lineMDLD/affine line
over
R
{\displaystyle {}\mathbb {R} }
connecting arrow MDLD/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 ,MDLD/vector space
U
⊆
V
{\displaystyle {}U\subseteq V}
linear subspace ,MDLD/linear subspace
and let
E
=
P
+
U
{\displaystyle {}E=P+U}
affine subspace .MDLD/affine subspace (translated)
Show that, for every point
Q
∈
E
{\displaystyle {}Q\in E}
E
=
Q
+
U
{\displaystyle {}E=Q+U}
Let
V
{\displaystyle {}V}
vector space ,MDLD/vector space
and let
E
⊆
V
{\displaystyle {}E\subseteq V}
affine subspace .MDLD/affine subspace (translated)
Show that
E
{\displaystyle {}E}
linear subspace MDLD/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 pointMDLD/starting point
and a
translation space.MDLD/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 pointMDLD/starting point
and a
translating space.MDLD/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 .MDLD/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 MDLD/affine subspace (translated)
of
K
[
X
]
≤
d
{\displaystyle {}K[X]_{\leq d}}
dimension MDLD/dimension (affine space)
of
E
{\displaystyle {}E}
E
{\displaystyle {}E}
Exercise Exercise 29.10
change ===
Let
E
{\displaystyle {}E}
affine space MDLD/affine space
over the
K
{\displaystyle {}K}
vector space MDLD/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 MDLD/affine space
over any
K
{\displaystyle {}K}
vector space MDLD/vector space
V
{\displaystyle {}V}
Let
E
{\displaystyle {}E}
K
{\displaystyle {}K}
vector space MDLD/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 MDLD/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 MDLD/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.
Exercise Exercise 29.14
change ===
Show that, for a family
P
i
{\displaystyle {}P_{i}}
i
∈
I
{\displaystyle {}i\in I}
affine space MDLD/affine space
E
{\displaystyle {}E}
barycentric combination MDLD/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}
Exercise * Exercise 29.15
change ===
Let
P
{\displaystyle {}P}
affine space MDLD/affine space
E
{\displaystyle {}E}
V
{\displaystyle {}V}
barycentric combinations MDLD/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 ,MDLD/vector space
which we consider as an
affine space MDLD/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 MDLD/affine space
over the
K
{\displaystyle {}K}
vector space MDLD/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 .MDLD/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 MDLD/vector space
over
K
{\displaystyle {}K}
affine space .MDLD/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 .MDLD/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 MDLD/barycentric coordinates
of your favorite color in additive mixing?
Exercise Exercise 29.20
change ===
Let
E
{\displaystyle {}E}
affine space MDLD/affine space
over a
K
{\displaystyle {}K}
vector space MDLD/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 MDLD/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 MDLD/affine basis
of the space.
Imagine four points in the natural
3
{\displaystyle {}3}
affine basis MDLD/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 pointMDLD/starting point
and a
translating space.MDLD/translating space
Let
V
{\displaystyle {}V}
K
{\displaystyle {}K}
vector space .MDLD/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 MDLD/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 MDLD/affine basis
of
E
{\displaystyle {}E}
P
i
{\displaystyle {}P_{i}}
V
×
K
{\displaystyle {}V\times K}
MDLD/vector space basis
of
V
×
K
{\displaystyle {}V\times K}
P
∈
E
{\displaystyle {}P\in E}
barycentric coordinates MDLD/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 MDLD/affine spaces
over the
field MDLD/field
K
{\displaystyle {}K}
MDLD/product space (set finite)
E
×
F
{\displaystyle {}E\times F}
Let
E
{\displaystyle {}E}
F
{\displaystyle {}F}
affine spaces MDLD/affine spaces
over the
field MDLD/field
K
{\displaystyle {}K}
affine basis MDLD/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)
