הוכחה:
אם
U
{\displaystyle U}
0
→
∈
U
{\displaystyle {\vec {0}}\in U}
תכונות החילוף, הפילוג, הקיבוץ והנייטרליות לכפל בסקלר נובעות במישרין מקיומן ב-
V
{\displaystyle V}
0
→
∈
U
{\displaystyle {\vec {0}}\in U}
a
∈
U
{\displaystyle a\in U}
U
≠
∅
{\displaystyle U\neq \emptyset }
u
=
v
=
a
,
α
=
−
1
{\displaystyle u=v=a\ ,\ \alpha =-1}
0
→
=
a
+
(
−
a
)
=
u
+
α
v
∈
U
{\displaystyle {\vec {0}}=a+(-a)=u+\alpha v\in U}
u
=
0
→
,
v
=
a
,
α
=
−
1
{\displaystyle u={\vec {0}}\ ,\ v=a\ ,\ \alpha =-1}
−
a
=
u
+
α
v
∈
U
{\displaystyle -a=u+\alpha v\in U}
a
,
b
∈
U
,
β
∈
F
{\displaystyle a,b\in U,\beta \in \mathbb {F} }
u
=
a
,
v
=
b
,
α
=
1
{\displaystyle u=a\ ,\ v=b\ ,\ \alpha =1}
a
+
b
∈
U
{\displaystyle a+b\in U}
u
=
0
→
,
α
=
β
,
v
=
a
{\displaystyle u={\vec {0}}\ ,\ \alpha =\beta \ ,\ v=a}
β
a
∈
U
{\displaystyle \beta a\in U}
U
{\displaystyle U}
F
{\displaystyle \mathbb {F} }
V
{\displaystyle V}
לכל
F
{\displaystyle \mathbb {F} }
V
{\displaystyle V}
U
=
{
0
}
{\displaystyle U=\left\{0\right\}}
V
{\displaystyle V}
לכל
F
{\displaystyle \mathbb {F} }
V
{\displaystyle V}
U
=
V
{\displaystyle U=V}
לכל
F
{\displaystyle \mathbb {F} }
V
=
F
n
{\displaystyle V=\mathbb {F} ^{n}}
V
,
U
=
{
(
x
1
,
.
.
.
x
n
)
∈
V
∣
x
i
=
0
∀
1
≤
i
≤
n
}
{\displaystyle V,U=\left\{\left(x_{1},...x_{n}\right)\in V\mid x_{i}=0\,\,\forall 1\leq i\leq n\right\}}
0
{\displaystyle 0}
לכל
F
{\displaystyle \mathbb {F} }
V
=
F
[
[
x
]
]
{\displaystyle V=\mathbb {F} \left[\left[x\right]\right]}
U
=
F
[
x
]
{\displaystyle U=\mathbb {F} \left[x\right]}
U
=
F
[
x
]
⊂
F
[
[
x
]
]
=
V
{\displaystyle U=\mathbb {F} \left[x\right]\subset \mathbb {F} \left[\left[x\right]\right]=V}
U
⊂
V
{\displaystyle U\subset V}
עריכה
תהי מטריצה
A
∈
M
m
×
n
{\displaystyle A\in M_{m\times n}}
F
{\displaystyle \mathbb {F} }
u
,
v
∈
F
n
{\displaystyle u,v\in \mathbb {F} ^{n}}
c
∈
F
{\displaystyle c\in \mathbb {F} }
חוק הפילוג חל על כפל מטריצה בוקטורים:
A
(
u
+
v
)
=
A
u
+
A
v
∈
F
m
{\displaystyle A\left(u+v\right)=Au+Av\in \mathbb {F} ^{m}}
חוק החילוף
A
(
c
⋅
u
)
=
c
⋅
(
A
u
)
∈
F
m
{\displaystyle A\left(c\cdot u\right)=c\cdot \left(Au\right)\in \mathbb {F} ^{m}}
אזי כאשר
U
=
{
v
∈
F
n
|
A
v
=
0
}
{\displaystyle U=\left\{v\in \mathbb {F} ^{n}|Av=0\right\}}
מטריצה הומוגנית ) הוא תמיד תת מרחב של
F
n
{\displaystyle \mathbb {F} ^{n}}
0
∈
U
{\displaystyle 0\in U}
u
+
v
∈
U
{\displaystyle u+v\in U}
u
,
v
∈
U
{\displaystyle u,v\in U}
A
u
=
0
{\displaystyle Au=0}
A
v
=
0
{\displaystyle Av=0}
A
(
u
+
v
)
=
A
u
+
A
v
=
0
+
0
=
0
{\displaystyle A\left(u+v\right)=Au+Av=0+0=0}
c
u
∈
U
{\displaystyle cu\in U}
u
∈
U
{\displaystyle u\in U}
c
∈
F
{\displaystyle c\in \mathbb {F} }
A
u
=
0
{\displaystyle Au=0}
A
(
c
u
)
=
c
(
A
u
)
=
c
⋅
0
=
0
{\displaystyle A\left(cu\right)=c\left(Au\right)=c\cdot 0=0}
תהי
A
=
I
n
{\displaystyle A=I_{n}}
U
=
{
v
∈
F
n
∣
I
n
v
=
0
}
=
{
0
}
{\displaystyle U=\left\{v\in \mathbb {F} ^{n}\mid I_{n}v=0\right\}=\left\{0\right\}}
A
=
0
{\displaystyle A=0}
U
=
{
v
∈
F
n
∣
0
v
=
0
}
=
F
n
{\displaystyle U=\left\{v\in \mathbb {F} ^{n}\mid 0v=0\right\}=\mathbb {F} ^{n}}
יהי
A
=
[
0
…
1
]
{\displaystyle A={\begin{bmatrix}0&\dots &1\end{bmatrix}}}
U
=
{
v
=
[
v
1
⋮
v
n
]
∈
F
n
∣
[
0
…
1
]
[
v
1
⋮
v
n
]
=
0
}
=
{
[
v
1
⋮
v
n
]
∈
F
n
∣
v
n
=
0
}
{\displaystyle U=\left\{v={\begin{bmatrix}v_{1}\\\vdots \\v_{n}\end{bmatrix}}\in \mathbb {F} ^{n}\mid {\begin{bmatrix}0&\dots &1\end{bmatrix}}{\begin{bmatrix}v_{1}\\\vdots \\v_{n}\end{bmatrix}}=0\right\}=\left\{{\begin{bmatrix}v_{1}\\\vdots \\v_{n}\end{bmatrix}}\in \mathbb {F} ^{n}\mid v_{n}=0\right\}}
![{\displaystyle V=\mathbb {F} \left[\left[x\right]\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c9cb80fc4bd4a989a9216373c788589d06ae6538)
![{\displaystyle U=\mathbb {F} \left[x\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2f88c71119e0c05b25341ea90455e24badf6def0)
![{\displaystyle U=\mathbb {F} \left[x\right]\subset \mathbb {F} \left[\left[x\right]\right]=V}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cd9484f0b0689508051fda22b2e876cfcf34afc1)
![{\displaystyle S=\left\{\left[{\begin{array}{c}x\\y\end{array}}\right]\in \mathbb {R} ^{2}|\ xy=0\right\}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/acbc44ff71cb9df2b9e9bd454c301d14f2edbf38)
![{\displaystyle \left[{\begin{array}{c}0\\0\end{array}}\right]\in S}](https://wikimedia.org/api/rest_v1/media/math/render/svg/82e2ee5c0fb7fdc2fa252404d78552399f9178de)
![{\displaystyle \left[{\begin{array}{c}1\\0\end{array}}\right],\left[{\begin{array}{c}0\\1\end{array}}\right]\in S}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3112f757599a0e7472ca51bd2d5fa470061be2cd)
![{\displaystyle v=\mathbb {R} \left[x\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4aabf3c1e162f78380f9e2c4757e831ba55f685e)
![{\displaystyle S=\left\{p(x)\in \mathbb {R} \left[x\right]|deg(p(x))\leq 2\land \ {\mbox{sum of p(x) is a rational number}}\right\}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a90d5b6dbe1af23358cbab233553fd0a94f9054f)
