מתוך ויקיספר, אוסף הספרים והמדריכים החופשי
הגדרה 1: טור
F
[
[
x
]
]
=
{
a
0
+
a
1
x
+
a
2
x
2
+
a
3
x
3
+
.
.
.
|
a
i
∈
F
}
{\displaystyle \mathbb {F} \left[\left[x\right]\right]=\left\{a_{0}+a_{1}x+a_{2}x^{2}+a_{3}x^{3}+...|a_{i}\in \mathbb {F} \right\}}
יהי
F
{\displaystyle \mathbb {F} }
V
=
F
[
[
x
]
]
{\displaystyle V=\mathbb {F} \left[\left[x\right]\right]}
טור הוא סכום אינסופי של חזקות. נסמן
S
=
{
1
,
x
,
x
2
,
.
.
.
}
∈
V
{\displaystyle S=\left\{1,x,x^{2},...\right\}\in V}
תרגיל 1:
s
p
a
n
(
S
)
=
F
[
[
x
]
]
{\displaystyle span\left(S\right)=\mathbb {F} \left[\left[x\right]\right]}
הגדרה 2: פולינום
F
[
x
]
=
{
a
0
+
a
1
x
+
.
.
.
a
n
x
n
|
n
∈
N
∪
{
0
}
,
a
i
∈
F
,
a
n
≠
0
}
∪
{
0
}
{\displaystyle \mathbb {F} \left[x\right]=\left\{a_{0}+a_{1}x+...a_{n}x^{n}|n\in \mathbb {N} \cup \left\{0\right\},\,a_{i}\in \mathbb {F} ,\,a_{n}\neq 0\right\}\cup \left\{0\right\}}
תרגיל 2:
V
=
F
[
x
]
{\displaystyle V=\mathbb {F} \left[x\right]}
S
=
{
1
,
x
,
x
2
,
.
.
.
}
∈
V
{\displaystyle S=\left\{1,x,x^{2},...\right\}\in V}
F
[
x
]
{\displaystyle \mathbb {F} \left[x\right]}
s
p
a
n
(
S
)
=
F
[
x
]
{\displaystyle span\left(S\right)=\mathbb {F} \left[x\right]}
S
{\displaystyle S}
כן. מפני שכל פולינום הוא סכום סופי של מחוברים מקבוצה של
S
{\displaystyle S}
0
≠
p
∈
F
[
x
]
{\displaystyle 0\neq p\in \mathbb {F} \left[x\right]}
a
0
,
.
.
,
a
n
∈
F
{\displaystyle a_{0},..,a_{n}\in \mathbb {F} }
p
=
a
0
+
a
1
x
+
.
.
.
a
n
x
n
∈
s
p
a
n
(
1
,
x
,
.
.
,
x
n
)
{\displaystyle p=a_{0}+a_{1}x+...a_{n}x^{n}\in span\left(1,x,..,x^{n}\right)}
s
p
a
n
(
S
)
{\displaystyle span\left(S\right)}
S
{\displaystyle S}
S
{\displaystyle S}
c
1
x
i
1
+
.
.
+
c
n
x
i
n
=
0
{\displaystyle c_{1}x^{i_{1}}+..+c_{n}x^{i_{n}}=0}
i
1
<
.
.
.
<
i
n
∈
N
∪
{
0
}
{\displaystyle i_{1}<...<i_{n}\in \mathbb {N} \cup \left\{0\right\}}
נניח בשלילה שקיים
1
≤
i
≤
n
{\displaystyle 1\leq i\leq n}
c
i
≠
0
{\displaystyle c_{i}\neq 0}
לכן
c
1
=
.
.
.
=
c
n
=
0
{\displaystyle c_{1}=...=c_{n}=0}
S
{\displaystyle S}
דוגמה 1: תלות לינארית של פולינום
יהיה
B
=
(
2
+
x
,
2
x
+
x
2
,
x
2
)
{\displaystyle B=(2+x,2x+x^{2},x^{2})}
B
{\displaystyle B}
a
(
2
+
x
)
+
b
(
2
x
+
x
2
)
+
c
x
2
=
0
{\displaystyle a(2+x)+b(2x+x^{2})+cx^{2}=0}
2
a
+
(
a
+
2
b
)
x
+
(
b
+
c
)
x
2
=
0
{\displaystyle 2a+(a+2b)x+(b+c)x^{2}=0}
אז הפולינום מתאפס אם ורק אם מקדמיו שווים לאפס:
a
=
0
,
a
+
2
b
=
0
,
b
+
c
=
0
{\displaystyle a=0,\ \ \ a+2b=0,\ \ \ b+c=0}
מ"ו של
F
[
x
]
{\displaystyle \mathbb {F} \left[x\right]}
F
[
[
x
]
]
{\displaystyle \mathbb {F} \left[\left[x\right]\right]}
(
v
1
,
.
.
,
v
n
)
{\displaystyle \left(v_{1},..,v_{n}\right)}
F
[
x
]
{\displaystyle \mathbb {F} \left[x\right]}
n
{\displaystyle n}
S
{\displaystyle S}
(
1
,
x
,
x
2
,
.
.
.
)
{\displaystyle \left(1,x,x^{2},...\right)}
S
{\displaystyle S}
F
[
x
]
{\displaystyle \mathbb {F} \left[x\right]}
לכל
n
∈
N
∪
{
0
}
{\displaystyle n\in \mathbb {N} \cup \left\{0\right\}}
F
[
x
]
≤
n
=
s
p
a
n
(
1
,
.
.
,
x
n
)
{\displaystyle \mathbb {F} \left[x\right]_{\leq n}=span\left(1,..,x^{n}\right)}
dim
(
F
[
x
]
≤
n
)
=
n
+
1
{\displaystyle \dim \left(\mathbb {F} \left[x\right]_{\leq n}\right)=n+1}
(
1
,
x
,
.
.
,
x
n
)
{\displaystyle \left(1,x,..,x^{n}\right)}
F
[
x
]
≤
n
{\displaystyle \mathbb {F} \left[x\right]_{\leq n}}
אם
a
0
+
a
1
x
+
.
.
+
a
k
x
k
∈
F
[
x
]
≤
n
{\displaystyle a_{0}+a_{1}x+..+a_{k}x^{k}\in \mathbb {F} \left[x\right]_{\leq n}}
a
k
≠
0
{\displaystyle a_{k}\neq 0}
k
≤
n
{\displaystyle k\leq n}
[
a
0
+
a
1
x
+
⋯
+
a
k
x
k
]
B
=
[
a
0
a
1
⋮
a
k
0
⋮
0
]
∈
F
n
+
1
{\displaystyle \left[a_{0}+a_{1}x+\cdots +a_{k}x^{k}\right]_{B}={\begin{bmatrix}a_{0}\\a_{1}\\\vdots \\a_{k}\\0\\\vdots \\0\end{bmatrix}}\in \mathbb {F} ^{n+1}}
![{\displaystyle \mathbb {F} \left[\left[x\right]\right]=\left\{a_{0}+a_{1}x+a_{2}x^{2}+a_{3}x^{3}+...|a_{i}\in \mathbb {F} \right\}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2469052d52b28747e79ca4429e1201bd7ab359c7)
![{\displaystyle V=\mathbb {F} \left[\left[x\right]\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c9cb80fc4bd4a989a9216373c788589d06ae6538)
![{\displaystyle span\left(S\right)=\mathbb {F} \left[\left[x\right]\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/33801208f5a8d23e2a2c202226cb229127fb8c73)
![{\displaystyle v=1+x+x^{2}+x^{3}+x^{4}+...\in \mathbb {F} \left[\left[x\right]\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c4526db0de7c397c99a149c8a0c157ef8be0fbb4)
![{\displaystyle \mathbb {F} \left[x\right]=\left\{a_{0}+a_{1}x+...a_{n}x^{n}|n\in \mathbb {N} \cup \left\{0\right\},\,a_{i}\in \mathbb {F} ,\,a_{n}\neq 0\right\}\cup \left\{0\right\}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/477504358b628f460248eb82ae80d8789f6cee6c)
![{\displaystyle V=\mathbb {F} \left[x\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cac0e413b50d5872d96522ea9f324cbf1721fc15)
![{\displaystyle \mathbb {F} \left[x\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6558e9a049e0817350acf4011a8a50bcaff44e2b)
![{\displaystyle span\left(S\right)=\mathbb {F} \left[x\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6d54a8772b9a2b3d16c4abc0ab2513509995254a)
![{\displaystyle 0\neq p\in \mathbb {F} \left[x\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ea051fa82b1f1ee2920783467a06a83784e17e13)
![{\displaystyle \mathbb {F} \left[\left[x\right]\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/af40e42acb495e73ef4a07a942c823e12de80299)
![{\displaystyle \mathbb {F} \left[x\right]_{\leq n}=span\left(1,..,x^{n}\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/874c872980cb76a558cfe25c2af92225e1558fe0)
![{\displaystyle \dim \left(\mathbb {F} \left[x\right]_{\leq n}\right)=n+1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/994720a2963e25a20fd20fd75d7454fcab300114)
![{\displaystyle \mathbb {F} \left[x\right]_{\leq n}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/99fdc6a665e1e662d46eee493e04d656bcc29c96)
![{\displaystyle a_{0}+a_{1}x+..+a_{k}x^{k}\in \mathbb {F} \left[x\right]_{\leq n}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/62df8c5e96f6ca763fab48c81537c96701db396e)
![{\displaystyle \left[a_{0}+a_{1}x+\cdots +a_{k}x^{k}\right]_{B}={\begin{bmatrix}a_{0}\\a_{1}\\\vdots \\a_{k}\\0\\\vdots \\0\end{bmatrix}}\in \mathbb {F} ^{n+1}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/78d238ac374775cfd9805ea231921194caf9724a)