מתוך ויקיספר, אוסף הספרים והמדריכים החופשי
הגדרה 1.8.1: 12 מטריצה מייצגת - מטריצת מעבר מבסיס B לבסיס C
נגדיר את קיום העתקה
T
{\displaystyle T}
יהיו
V
,
W
{\displaystyle V,W}
F
{\displaystyle \mathbb {F} }
B
=
(
v
1
,
…
,
v
n
)
,
C
=
(
w
1
,
⋯
,
w
m
)
{\displaystyle B=(v_{1},\ldots ,v_{n}),C=(w_{1},\cdots ,w_{m})}
V
,
W
{\displaystyle V,W}
V
,
W
{\displaystyle V,W}
n
,
m
{\displaystyle n,m}
T
:
V
→
W
{\displaystyle T:V\to W}
T
{\displaystyle T}
B
,
C
{\displaystyle B,C}
A
{\displaystyle A}
j
{\displaystyle j}
A
{\displaystyle A}
[
T
(
v
j
)
]
C
{\displaystyle {\bigl [}T(v_{j}){\bigr ]}_{C}}
טענה 1: קיום מטריצה
T
{\displaystyle T}
יהיו
V
,
W
{\displaystyle V,W}
F
{\displaystyle \mathbb {F} }
B
=
(
v
1
,
…
,
v
n
)
,
C
=
(
w
1
,
⋯
,
w
m
)
{\displaystyle B=(v_{1},\ldots ,v_{n}),C=(w_{1},\cdots ,w_{m})}
V
,
W
{\displaystyle V,W}
A
∈
M
m
×
n
{\displaystyle A\in M_{m\times n}}
F
{\displaystyle \mathbb {F} }
T
:
V
→
W
{\displaystyle T:V\to W}
[
T
]
C
B
=
A
{\displaystyle [T]_{C}^{B}=A}
הוכחה: נסמן את המקדמים של
A
{\displaystyle A}
a
i
j
{\displaystyle a_{ij}}
i
{\displaystyle i}
j
{\displaystyle j}
A
=
[
a
11
⋯
a
1
n
⋮
⋮
a
m
1
⋯
a
m
n
]
{\displaystyle A={\begin{bmatrix}a_{11}&\cdots &a_{1n}\\\vdots &&\vdots \\a_{m1}&\cdots &a_{mn}\end{bmatrix}}}
אז
[
T
]
C
B
=
A
{\displaystyle [T]_{C}^{B}=A}
1
≤
j
≤
n
{\displaystyle 1\leq j\leq n}
[
T
(
v
j
)
]
C
=
[
a
1
j
⋮
a
m
j
]
{\displaystyle {\bigl [}T(v_{j}){\bigr ]}_{C}={\begin{bmatrix}a_{1j}\\\vdots \\a_{mj}\end{bmatrix}}}
T
{\displaystyle T}
B
{\displaystyle B}
כל וקטור
w
∈
W
{\displaystyle w\in W}
W
{\displaystyle W}
w
=
a
1
w
1
+
⋯
+
a
m
w
m
{\displaystyle w=a_{1}w_{1}+\cdots +a_{m}w_{m}}
אם אנחנו מפעילים העתקה על וקטורי
B
{\displaystyle B}
C
{\displaystyle C}
[
T
]
C
B
=
A
{\displaystyle [T]_{C}^{B}=A}
זה מתקיים אמ"מ:
T
(
v
j
)
=
a
1
j
w
1
+
.
.
.
+
a
m
j
w
m
{\displaystyle T\left(v_{j}\right)=a_{1j}w_{1}+...+a_{mj}w_{m}}
לפי המשפט קיום ויחידות ה"ל, קיימת
T
:
V
→
W
{\displaystyle T:V\to W}
T
(
v
j
)
=
a
1
j
w
1
+
⋯
+
a
m
j
w
m
{\displaystyle T(v_{j})=a_{1j}w_{1}+\cdots +a_{mj}w_{m}}
[
T
]
C
B
=
A
{\displaystyle [T]_{C}^{B}=A}
משפט 1:
[
T
]
C
B
⋅
[
v
]
B
=
[
T
(
v
)
]
C
=
A
[
v
]
B
{\displaystyle [T]_{C}^{B}\cdot [v]_{B}={\bigl [}T(v){\bigr ]}_{C}=A[v]_{B}}
B
{\displaystyle B}
C
{\displaystyle C}
יהיו
V
,
W
{\displaystyle V,W}
F
{\displaystyle \mathbb {F} }
B
=
(
v
1
,
…
,
v
n
)
,
C
=
(
w
1
,
⋯
,
w
m
)
{\displaystyle B=(v_{1},\ldots ,v_{n}),C=(w_{1},\cdots ,w_{m})}
V
,
W
{\displaystyle V,W}
v
∈
V
{\displaystyle v\in V}
[
T
]
C
B
⋅
[
v
]
B
=
[
T
(
v
)
]
C
{\displaystyle [T]_{C}^{B}\cdot [v]_{B}={\bigl [}T(v){\bigr ]}_{C}}
הוכחה:
נסמן
[
v
]
B
=
[
c
1
⋮
c
n
]
,
[
T
]
C
B
=
(
a
11
⋯
a
1
n
⋮
⋮
a
m
1
⋯
a
m
n
)
{\displaystyle [v]_{B}={\begin{bmatrix}c_{1}\\\vdots \\c_{n}\end{bmatrix}},\quad [T]_{C}^{B}={\begin{pmatrix}a_{11}&\cdots &a_{1n}\\\vdots &&\vdots \\a_{m1}&\cdots &a_{mn}\end{pmatrix}}}
B
{\displaystyle B}
v
=
c
1
v
1
+
⋯
+
c
n
v
n
=
∑
j
=
1
m
n
c
j
v
j
∗
{\displaystyle v=c_{1}v_{1}+\cdots +c_{n}v_{n}=\sum _{j=1}^{m}nc_{j}v_{j}*}
בנוסף מהמטריצה
T
{\displaystyle T}
[
T
(
v
j
)
]
C
=
[
a
i
j
⋮
a
m
j
]
{\displaystyle {\bigl [}T(v_{j}){\bigr ]}_{C}={\begin{bmatrix}a_{ij}\\\vdots \\a_{mj}\end{bmatrix}}}
[
T
(
v
j
)
]
C
=
a
1
j
w
1
+
⋯
+
a
m
j
w
m
=
∑
i
=
1
m
m
a
i
j
w
i
{\displaystyle {\bigl [}T(v_{j}){\bigr ]}_{C}=a_{1j}w_{1}+\cdots +a_{mj}w_{m}=\sum _{i=1}^{m}ma_{ij}w_{i}}
נבצע העתקה על
v
∗
{\displaystyle v*}
T
(
v
)
=
T
(
∑
j
=
1
m
n
c
j
v
j
)
=
∑
j
=
1
m
n
c
j
T
(
v
j
)
=
∑
j
=
1
m
n
c
j
(
∑
j
=
1
m
m
a
i
j
w
i
)
=
⏞
∗
∑
j
=
1
m
n
∑
j
=
1
m
m
a
i
j
⋅
c
j
⋅
w
i
=
∑
j
=
1
m
m
(
∑
j
=
1
m
n
a
i
j
⋅
c
j
)
⋅
w
i
{\displaystyle T(v)=T\left(\sum _{j=1}^{m}nc_{j}v_{j}\right)=\sum _{j=1}^{m}nc_{j}T(v_{j})=\sum _{j=1}^{m}nc_{j}\left(\sum _{j=1}^{m}ma_{ij}w_{i}\right)\overbrace {=} ^{*}\sum _{j=1}^{m}n\sum _{j=1}^{m}ma_{ij}\cdot c_{j}\cdot w_{i}=\sum _{j=1}^{m}m\left(\sum _{j=1}^{m}na_{ij}\cdot c_{j}\right)\cdot w_{i}}
מתכונות ה"ל, ואסוציאטיביות (החלפת סדר מחוברים) נסמן
b
i
=
∑
j
=
1
m
n
a
i
j
c
j
{\displaystyle b_{i}=\sum _{j=1}^{m}na_{ij}c_{j}}
[
T
(
v
j
)
]
C
=
∑
i
=
1
m
m
b
i
w
i
{\displaystyle {\bigl [}T(v_{j}){\bigr ]}_{C}=\sum _{i=1}^{m}mb_{i}w_{i}}
[
T
(
v
)
]
C
=
(
b
1
⋮
b
m
)
{\displaystyle {\bigl [}T(v){\bigr ]}_{C}={\begin{pmatrix}b_{1}\\\vdots \\b_{m}\end{pmatrix}}}
בנוסף
[
T
]
C
B
[
v
]
B
=
[
a
11
⋯
a
1
n
⋮
⋮
a
m
1
⋯
a
m
n
]
[
c
1
⋮
c
n
]
=
[
b
1
⋮
b
m
]
=
[
T
(
v
)
]
C
{\displaystyle [T]_{C}^{B}[v]_{B}={\begin{bmatrix}a_{11}&\cdots &a_{1n}\\\vdots &&\vdots \\a_{m1}&\cdots &a_{mn}\end{bmatrix}}{\begin{bmatrix}c_{1}\\\vdots \\c_{n}\end{bmatrix}}={\begin{bmatrix}b_{1}\\\vdots \\b_{m}\end{bmatrix}}={\bigl [}T(v){\bigr ]}_{C}}
מסקנה: בהינתן מ"ו עם בסיס סדור
B
=
(
v
1
,
…
,
v
n
)
{\displaystyle B=(v_{1},\ldots ,v_{n})}
T
B
:
F
n
→
V
{\displaystyle T_{B}:\mathbb {F} ^{n}\to V}
T
B
[
c
1
⋮
c
n
]
=
c
1
v
1
+
⋯
+
c
n
v
n
{\displaystyle T_{B}{\begin{bmatrix}c_{1}\\\vdots \\c_{n}\end{bmatrix}}=c_{1}v_{1}+\cdots +c_{n}v_{n}}
S
B
(
v
)
=
[
v
]
B
{\displaystyle S_{B}(v)=[v]_{B}}
S
B
:
V
→
F
n
{\displaystyle S_{B}:V\to \mathbb {F} ^{n}}
T
B
,
S
B
{\displaystyle T_{B},S_{B}}
V
→
⏞
S
B
F
n
{\displaystyle V\overbrace {\to } ^{S_{B}}\mathbb {F} ^{n}}
W
→
⏞
S
c
F
m
{\displaystyle W\overbrace {\to } ^{S_{c}}\mathbb {F} ^{m}}
V
,
W
{\displaystyle V,W}
F
{\displaystyle \mathbb {F} }
B
,
C
{\displaystyle B,C}
V
,
W
{\displaystyle V,W}
תהי
T
:
V
→
W
{\displaystyle T:V\to W}
A
=
[
T
]
C
B
{\displaystyle A=[T]_{C}^{B}}
T
A
(
x
)
=
A
x
T
A
:
F
n
→
F
m
{\displaystyle T_{A}(x)=Ax\quad T_{A}:\mathbb {F} ^{n}\to \mathbb {F} ^{m}}
![{\displaystyle {\bigl [}T(v_{j}){\bigr ]}_{C}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7a44867fe9290e47e583450b91e35774010550f3)
![{\displaystyle [T]_{C}^{B}=A}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9535ff9f86bdfee01ed13838f247573f96e853ab)
![{\displaystyle {\bigl [}T(v_{j}){\bigr ]}_{C}={\begin{bmatrix}a_{1j}\\\vdots \\a_{mj}\end{bmatrix}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6b9bb69d5830da3498354eeee9832659e1bc17a4)
![{\displaystyle [T]_{C}^{B}\cdot [v]_{B}={\bigl [}T(v){\bigr ]}_{C}=A[v]_{B}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bb93e655f5163e3932866d228c344fabe4eb9846)
![{\displaystyle [T]_{C}^{B}\cdot [v]_{B}={\bigl [}T(v){\bigr ]}_{C}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3200946b7c6b4a11ab88663dfe8f08724a51e0f7)
![{\displaystyle [v]_{B}={\begin{bmatrix}c_{1}\\\vdots \\c_{n}\end{bmatrix}},\quad [T]_{C}^{B}={\begin{pmatrix}a_{11}&\cdots &a_{1n}\\\vdots &&\vdots \\a_{m1}&\cdots &a_{mn}\end{pmatrix}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a2a60ce0dc4e409d3878da742d6dbc523fa81a17)
![{\displaystyle {\bigl [}T(v_{j}){\bigr ]}_{C}={\begin{bmatrix}a_{ij}\\\vdots \\a_{mj}\end{bmatrix}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8bc814bf35651f1e9291a0c3d1bcda38f7241195)
![{\displaystyle {\bigl [}T(v_{j}){\bigr ]}_{C}=a_{1j}w_{1}+\cdots +a_{mj}w_{m}=\sum _{i=1}^{m}ma_{ij}w_{i}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1c18479ad6e469fd1939c3127e4bbe2c2eb1d654)
![{\displaystyle {\bigl [}T(v_{j}){\bigr ]}_{C}=\sum _{i=1}^{m}mb_{i}w_{i}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2e4b139be9c4e8fbbb5158e67aeed1e4d7b7e311)
![{\displaystyle {\bigl [}T(v){\bigr ]}_{C}={\begin{pmatrix}b_{1}\\\vdots \\b_{m}\end{pmatrix}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d46d6ffcacdae672c13b635f9098110294b548a2)
![{\displaystyle [T]_{C}^{B}[v]_{B}={\begin{bmatrix}a_{11}&\cdots &a_{1n}\\\vdots &&\vdots \\a_{m1}&\cdots &a_{mn}\end{bmatrix}}{\begin{bmatrix}c_{1}\\\vdots \\c_{n}\end{bmatrix}}={\begin{bmatrix}b_{1}\\\vdots \\b_{m}\end{bmatrix}}={\bigl [}T(v){\bigr ]}_{C}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d7ca39b89d2efa5cc685ea437d993a6414ef3320)
![{\displaystyle S_{B}(v)=[v]_{B}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b34984ebc657f9d00b03aef12e0c199a956375c9)
![{\displaystyle A=[T]_{C}^{B}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/206e7239d17cef8d18f44968828b9d1a6be6ea0d)
![{\displaystyle (S_{C}\circ T)(v)=S_{C}{\bigl (}T(v){\bigr )}={\bigl [}T(v){\bigr ]}_{C}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d0e4d75ff97b634161a6c21a82dc40c531177c21)
![{\displaystyle (T_{A}\circ S_{B})(v)=T_{A}{\bigl (}S_{B}(v){\bigr )}=T_{A}([v]_{B})=A\cdot [v]_{B}=[T]_{C}^{B}[v]_{B}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d78749e2150bbf3ed0c57545a5b02bb921af67ec)