נמצא את <
[
T
A
]
E
B
{\displaystyle [T_{A}]_{E}^{B}}
ראשית נמצא את ייצוגו של הווקטור לאחר הפעלה של העתקה לינארית עליו:
T
A
(
v
1
)
=
[
1
0
0
−
1
]
[
1
1
]
=
[
1
−
1
]
{\displaystyle T_{A}(v_{1})=\left[{\begin{matrix}1&0\\0&-1\end{matrix}}\right]\left[{\begin{matrix}1\\1\end{matrix}}\right]=\left[{\begin{matrix}1\\-1\end{matrix}}\right]}
T
A
(
v
2
)
=
[
1
0
0
−
1
]
[
2
1
]
=
[
2
−
1
]
{\displaystyle T_{A}(v_{2})=\left[{\begin{matrix}1&0\\0&-1\end{matrix}}\right]\left[{\begin{matrix}2\\1\end{matrix}}\right]=\left[{\begin{matrix}2\\-1\end{matrix}}\right]}
לכן בכדי למצוא את
[
T
A
]
E
B
{\displaystyle [T_{A}]_{E}^{B}}
B
{\displaystyle B}
E
{\displaystyle E}
R
2
{\displaystyle R^{2}}
E
{\displaystyle E}
במילים אחרות עלינו לשאול באיזה סקלר הוכפלו ווקטורי הבסיס
E
=
{
[
1
0
]
,
[
0
1
]
}
{\displaystyle E={\Biggl \{}\left[{\begin{matrix}1\\0\end{matrix}}\right],\left[{\begin{matrix}0\\1\end{matrix}}\right]{\Biggl \}}}
T
A
(
v
1
)
=
[
1
−
1
]
{\displaystyle T_{A}(v_{1})=\left[{\begin{matrix}1\\-1\end{matrix}}\right]}
T
A
(
v
2
)
=
[
2
−
1
]
{\displaystyle T_{A}(v_{2})=\left[{\begin{matrix}2\\-1\end{matrix}}\right]}
x
11
∗
[
1
0
]
+
x
21
∗
[
0
1
]
=
[
1
1
]
{\displaystyle x_{11}*\left[{\begin{matrix}1\\0\end{matrix}}\right]+x_{21}*\left[{\begin{matrix}0\\1\end{matrix}}\right]=\left[{\begin{matrix}1\\1\end{matrix}}\right]}
x
12
∗
[
1
0
]
+
x
22
∗
[
0
1
]
=
[
2
−
1
]
{\displaystyle x_{12}*\left[{\begin{matrix}1\\0\end{matrix}}\right]+x_{22}*\left[{\begin{matrix}0\\1\end{matrix}}\right]=\left[{\begin{matrix}2\\-1\end{matrix}}\right]}
נפתור את מערכת המשוואות ונקבל כי
[
x
11
x
21
]
=
[
1
1
]
,
[
x
12
x
22
]
=
[
2
−
1
]
{\displaystyle \left[{\begin{matrix}x_{11}\\x_{21}\end{matrix}}\right]=\left[{\begin{matrix}1\\1\end{matrix}}\right],\left[{\begin{matrix}x_{12}\\x_{22}\end{matrix}}\right]=\left[{\begin{matrix}2\\-1\end{matrix}}\right]}
על כן המטריצה שלנו הינה:
[
T
A
]
E
B
=
[
1
2
1
−
1
]
{\displaystyle [T_{A}]_{E}^{B}=\left[{\begin{matrix}1&2\\1&-1\end{matrix}}\right]}
נמצא את <
[
T
A
]
B
B
{\displaystyle [T_{A}]_{B}^{B}}
ראשית נמצא את ייצוגו של הווקטור לאחר הפעלה של העתקה לינארית עליו:
T
A
(
v
1
)
=
[
1
0
0
−
1
]
[
1
1
]
=
[
1
−
1
]
{\displaystyle T_{A}(v_{1})=\left[{\begin{matrix}1&0\\0&-1\end{matrix}}\right]\left[{\begin{matrix}1\\1\end{matrix}}\right]=\left[{\begin{matrix}1\\-1\end{matrix}}\right]}
T
A
(
v
2
)
=
[
1
0
0
−
1
]
[
2
1
]
=
[
2
−
1
]
{\displaystyle T_{A}(v_{2})=\left[{\begin{matrix}1&0\\0&-1\end{matrix}}\right]\left[{\begin{matrix}2\\1\end{matrix}}\right]=\left[{\begin{matrix}2\\-1\end{matrix}}\right]}
לכן בכדי למצוא את
[
T
A
]
B
B
{\displaystyle [T_{A}]_{B}^{B}}
B
{\displaystyle B}
B
{\displaystyle B}
R
2
{\displaystyle R^{2}}
B
{\displaystyle B}
במילים אחרות עלינו לשאול באיזה סקלר הוכפלו ווקטורי הבסיס
B
=
(
v
1
,
v
2
)
=
{
[
1
1
]
,
[
2
1
]
}
{\displaystyle B=(v_{1},v_{2})={\Biggl \{}\left[{\begin{matrix}1\\1\end{matrix}}\right],\left[{\begin{matrix}2\\1\end{matrix}}\right]{\Biggl \}}}
T
A
(
v
1
)
=
[
1
−
1
]
{\displaystyle T_{A}(v_{1})=\left[{\begin{matrix}1\\-1\end{matrix}}\right]}
T
A
(
v
1
)
=
[
2
−
1
]
{\displaystyle T_{A}(v_{1})=\left[{\begin{matrix}2\\-1\end{matrix}}\right]}
x
11
∗
[
1
1
]
+
x
21
∗
[
2
1
]
=
[
1
−
1
]
{\displaystyle x_{11}*\left[{\begin{matrix}1\\1\end{matrix}}\right]+x_{21}*\left[{\begin{matrix}2\\1\end{matrix}}\right]=\left[{\begin{matrix}1\\-1\end{matrix}}\right]}
x
12
∗
[
1
1
]
+
x
22
∗
[
2
1
]
=
[
2
−
1
]
{\displaystyle x_{12}*\left[{\begin{matrix}1\\1\end{matrix}}\right]+x_{22}*\left[{\begin{matrix}2\\1\end{matrix}}\right]=\left[{\begin{matrix}2\\-1\end{matrix}}\right]}
נפתור את מערכת המשוואות ונקבל כי
[
x
11
x
21
]
=
[
−
1
1
]
,
[
x
12
x
22
]
=
[
0
1
]
{\displaystyle \left[{\begin{matrix}x_{11}\\x_{21}\end{matrix}}\right]=\left[{\begin{matrix}-1\\1\end{matrix}}\right],\left[{\begin{matrix}x_{12}\\x_{22}\end{matrix}}\right]=\left[{\begin{matrix}0\\1\end{matrix}}\right]}
על כן המטריצה שלנו הינה:
[
T
A
]
B
B
=
[
−
1
0
1
1
]
{\displaystyle [T_{A}]_{B}^{B}=\left[{\begin{matrix}-1&0\\1&1\end{matrix}}\right]}
![{\displaystyle {\color {red}[T(v)]_{c}}=[T]_{c}^{b}*[v]_{b}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/eb905923f398defd7244cd3b9f4459e0063b73b0)
![{\displaystyle B=(v_{1},v_{2})={\Biggl \{}\left[{\begin{matrix}1\\1\end{matrix}}\right],\left[{\begin{matrix}2\\1\end{matrix}}\right]{\Biggl \}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3a092d32718b5d2b3675110fe4841b8ce74a6acb)
![{\displaystyle A={\Biggl \{}\left[{\begin{matrix}1&0\\0&-1\end{matrix}}\right]{\Biggl \}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6f4b4c87dbfbe8883832891b6836fee00d155c81)
![{\displaystyle [T_{A}]_{E}^{B}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f87389ba5485dfa3584422872b1821de95c9f8cb)
![{\displaystyle T_{A}(v_{1})=\left[{\begin{matrix}1&0\\0&-1\end{matrix}}\right]\left[{\begin{matrix}1\\1\end{matrix}}\right]=\left[{\begin{matrix}1\\-1\end{matrix}}\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/124342603bff13e21351a1e3f0d512a8c995953a)
![{\displaystyle T_{A}(v_{2})=\left[{\begin{matrix}1&0\\0&-1\end{matrix}}\right]\left[{\begin{matrix}2\\1\end{matrix}}\right]=\left[{\begin{matrix}2\\-1\end{matrix}}\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6c1a54d24d566809004da5d572d1d9dfffbc8d82)
![{\displaystyle E={\Biggl \{}\left[{\begin{matrix}1\\0\end{matrix}}\right],\left[{\begin{matrix}0\\1\end{matrix}}\right]{\Biggl \}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/225aca46caf13c849afcc34ad65e4979b3e5aab3)
![{\displaystyle T_{A}(v_{1})=\left[{\begin{matrix}1\\-1\end{matrix}}\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1f7bf37535e271fb3ce3e58163a57e01b6a3eb45)
![{\displaystyle T_{A}(v_{2})=\left[{\begin{matrix}2\\-1\end{matrix}}\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/67de021a5626da86c2f1649a04ae62711d71173a)
![{\displaystyle x_{11}*\left[{\begin{matrix}1\\0\end{matrix}}\right]+x_{21}*\left[{\begin{matrix}0\\1\end{matrix}}\right]=\left[{\begin{matrix}1\\1\end{matrix}}\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0dc6bb8cedf9f7b6fa25027cb0a656af7abae5bb)
![{\displaystyle x_{12}*\left[{\begin{matrix}1\\0\end{matrix}}\right]+x_{22}*\left[{\begin{matrix}0\\1\end{matrix}}\right]=\left[{\begin{matrix}2\\-1\end{matrix}}\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7ced7608c7d2654ad3f3e77f5649089e812cbbf0)
![{\displaystyle \left[{\begin{matrix}x_{11}\\x_{21}\end{matrix}}\right]=\left[{\begin{matrix}1\\1\end{matrix}}\right],\left[{\begin{matrix}x_{12}\\x_{22}\end{matrix}}\right]=\left[{\begin{matrix}2\\-1\end{matrix}}\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/39baebf2e885d8ae9af47c6a2592f90897f407c4)
![{\displaystyle [T_{A}]_{E}^{B}=\left[{\begin{matrix}1&2\\1&-1\end{matrix}}\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cb83e3a5dd92f0e0880cad2b797a577bf0f2017f)
![{\displaystyle [T_{A}]_{B}^{B}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f8e832390b7abd95fe2c7acfef91275260884333)
![{\displaystyle T_{A}(v_{1})=\left[{\begin{matrix}2\\-1\end{matrix}}\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/dbc00543437e25bcc218c9d88a557252bf40a763)
![{\displaystyle x_{11}*\left[{\begin{matrix}1\\1\end{matrix}}\right]+x_{21}*\left[{\begin{matrix}2\\1\end{matrix}}\right]=\left[{\begin{matrix}1\\-1\end{matrix}}\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1c3e11911107ce0e5b01aa6eb0638d275f37c247)
![{\displaystyle x_{12}*\left[{\begin{matrix}1\\1\end{matrix}}\right]+x_{22}*\left[{\begin{matrix}2\\1\end{matrix}}\right]=\left[{\begin{matrix}2\\-1\end{matrix}}\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9afb787f5038835f0a6d2fca5749624ba6b44fb4)
![{\displaystyle \left[{\begin{matrix}x_{11}\\x_{21}\end{matrix}}\right]=\left[{\begin{matrix}-1\\1\end{matrix}}\right],\left[{\begin{matrix}x_{12}\\x_{22}\end{matrix}}\right]=\left[{\begin{matrix}0\\1\end{matrix}}\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/125e24a06ffe498f875294f78ed3ba9d108a7adc)
![{\displaystyle [T_{A}]_{B}^{B}=\left[{\begin{matrix}-1&0\\1&1\end{matrix}}\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a65697d0c9e8afb99281bd412fead82dc6bb623c)