Transformation matrix with 2 bases
$begingroup$
I have the following:
$ C = ($$ left[
begin{array}{cc}
1\
-1
end{array}
right], left[
begin{array}
-1\
2
end{array}
right]) ,
B = ($$ left[
begin{array}{cc}
2\
1
end{array}
right], left[
begin{array}{c}
3\
2
end{array}
right]) $
are bases of $mathbb{R}^2$ ( no need to prove).
$ T:mathbb{R}^2 to mathbb{R}^2 $ is linear transformation such that $ [T]^B _B = $$ left[
begin{array}{cc}
1&2\
-1&1
end{array}
right] $$ $
I need to calculate $ [T]^C _C$.
from $[T] ^B _B$ I can gather that:
$ T($$ left[
begin{array}{cc}
2\
1
end{array}
right]) = 1 * $$ left[
begin{array}{cc}
2\
1
end{array}
right] - 1* $$ left[
begin{array}{cc}
3\
2
end{array}
right] $
$ T($$ left[
begin{array}{cc}
3\
2
end{array}
right]) = 2 * $$ left[
begin{array}{cc}
2\
1
end{array}
right] + 1* $$ left[
begin{array}{cc}
3\
2
end{array}
right] $
And then I can know the values of those, but how to progress from here to find $[T]^C _C$?
linear-algebra matrices linear-transformations change-of-basis
asked Dec 27 '18 at 18:23
TegernakoTegernako
948
$endgroup$
add a comment |
$begingroup$
I have the following:
$ C = ($$ left[
begin{array}{cc}
1\
-1
end{array}
right], left[
begin{array}
-1\
2
end{array}
right]) ,
B = ($$ left[
begin{array}{cc}
2\
1
end{array}
right], left[
begin{array}{c}
3\
2
end{array}
right]) $
are bases of $mathbb{R}^2$ ( no need to prove).
$ T:mathbb{R}^2 to mathbb{R}^2 $ is linear transformation such that $ [T]^B _B = $$ left[
begin{array}{cc}
1&2\
-1&1
end{array}
right] $$ $
I need to calculate $ [T]^C _C$.
from $[T] ^B _B$ I can gather that:
$ T($$ left[
begin{array}{cc}
2\
1
end{array}
right]) = 1 * $$ left[
begin{array}{cc}
2\
1
end{array}
right] - 1* $$ left[
begin{array}{cc}
3\
2
end{array}
right] $
$ T($$ left[
begin{array}{cc}
3\
2
end{array}
right]) = 2 * $$ left[
begin{array}{cc}
2\
1
end{array}
right] + 1* $$ left[
begin{array}{cc}
3\
2
end{array}
right] $
And then I can know the values of those, but how to progress from here to find $[T]^C _C$?
linear-algebra matrices linear-transformations change-of-basis
asked Dec 27 '18 at 18:23
TegernakoTegernako
948
$endgroup$
1
$begingroup$
Find the matrix $M$ of change of basis from $B$ to $C$, and the matrix $N$ of change of basis from $C$ to $B$ (in fact, $N=M^{-1}$). Finally, $[T]_C^C=Ncdot [T]_B^Bcdot M$.
$endgroup$
– Tito Eliatron
Dec 27 '18 at 18:26
1
$begingroup$
just see the image of basis C using given transformation.
$endgroup$
– ASHWINI SANKHE
Dec 28 '18 at 7:20
$begingroup$
Thanks, we still did not learn that formula but were already given it in homework.. Have a good weekend!
$endgroup$
– Tegernako
Dec 28 '18 at 10:38
add a comment |
$begingroup$
I have the following:
$ C = ($$ left[
begin{array}{cc}
1\
-1
end{array}
right], left[
begin{array}
-1\
2
end{array}
right]) ,
B = ($$ left[
begin{array}{cc}
2\
1
end{array}
right], left[
begin{array}{c}
3\
2
end{array}
right]) $
are bases of $mathbb{R}^2$ ( no need to prove).
$ T:mathbb{R}^2 to mathbb{R}^2 $ is linear transformation such that $ [T]^B _B = $$ left[
begin{array}{cc}
1&2\
-1&1
end{array}
right] $$ $
I need to calculate $ [T]^C _C$.
from $[T] ^B _B$ I can gather that:
$ T($$ left[
begin{array}{cc}
2\
1
end{array}
right]) = 1 * $$ left[
begin{array}{cc}
2\
1
end{array}
right] - 1* $$ left[
begin{array}{cc}
3\
2
end{array}
right] $
$ T($$ left[
begin{array}{cc}
3\
2
end{array}
right]) = 2 * $$ left[
begin{array}{cc}
2\
1
end{array}
right] + 1* $$ left[
begin{array}{cc}
3\
2
end{array}
right] $
And then I can know the values of those, but how to progress from here to find $[T]^C _C$?
linear-algebra matrices linear-transformations change-of-basis
asked Dec 27 '18 at 18:23
TegernakoTegernako
948
$endgroup$
I have the following:
$ C = ($$ left[
begin{array}{cc}
1\
-1
end{array}
right], left[
begin{array}
-1\
2
end{array}
right]) ,
B = ($$ left[
begin{array}{cc}
2\
1
end{array}
right], left[
begin{array}{c}
3\
2
end{array}
right]) $
are bases of $mathbb{R}^2$ ( no need to prove).
$ T:mathbb{R}^2 to mathbb{R}^2 $ is linear transformation such that $ [T]^B _B = $$ left[
begin{array}{cc}
1&2\
-1&1
end{array}
right] $$ $
I need to calculate $ [T]^C _C$.
from $[T] ^B _B$ I can gather that:
$ T($$ left[
begin{array}{cc}
2\
1
end{array}
right]) = 1 * $$ left[
begin{array}{cc}
2\
1
end{array}
right] - 1* $$ left[
begin{array}{cc}
3\
2
end{array}
right] $
$ T($$ left[
begin{array}{cc}
3\
2
end{array}
right]) = 2 * $$ left[
begin{array}{cc}
2\
1
end{array}
right] + 1* $$ left[
begin{array}{cc}
3\
2
end{array}
right] $
And then I can know the values of those, but how to progress from here to find $[T]^C _C$?
linear-algebra matrices linear-transformations change-of-basis
linear-algebra matrices linear-transformations change-of-basis
asked Dec 27 '18 at 18:23
TegernakoTegernako
948
asked Dec 27 '18 at 18:23
TegernakoTegernako
948
asked Dec 27 '18 at 18:23
TegernakoTegernako
948
asked Dec 27 '18 at 18:23
TegernakoTegernako
948
asked Dec 27 '18 at 18:23
TegernakoTegernako
948
948
1
$begingroup$
Find the matrix $M$ of change of basis from $B$ to $C$, and the matrix $N$ of change of basis from $C$ to $B$ (in fact, $N=M^{-1}$). Finally, $[T]_C^C=Ncdot [T]_B^Bcdot M$.
$endgroup$
– Tito Eliatron
Dec 27 '18 at 18:26
1
$begingroup$
just see the image of basis C using given transformation.
$endgroup$
– ASHWINI SANKHE
Dec 28 '18 at 7:20
$begingroup$
Thanks, we still did not learn that formula but were already given it in homework.. Have a good weekend!
$endgroup$
– Tegernako
Dec 28 '18 at 10:38
add a comment |
1
$begingroup$
Find the matrix $M$ of change of basis from $B$ to $C$, and the matrix $N$ of change of basis from $C$ to $B$ (in fact, $N=M^{-1}$). Finally, $[T]_C^C=Ncdot [T]_B^Bcdot M$.
$endgroup$
– Tito Eliatron
Dec 27 '18 at 18:26
1
$begingroup$
just see the image of basis C using given transformation.
$endgroup$
– ASHWINI SANKHE
Dec 28 '18 at 7:20
$begingroup$
Thanks, we still did not learn that formula but were already given it in homework.. Have a good weekend!
$endgroup$
– Tegernako
Dec 28 '18 at 10:38
1
1
$begingroup$
Find the matrix $M$ of change of basis from $B$ to $C$, and the matrix $N$ of change of basis from $C$ to $B$ (in fact, $N=M^{-1}$). Finally, $[T]_C^C=Ncdot [T]_B^Bcdot M$.
$endgroup$
– Tito Eliatron
Dec 27 '18 at 18:26
$begingroup$
Find the matrix $M$ of change of basis from $B$ to $C$, and the matrix $N$ of change of basis from $C$ to $B$ (in fact, $N=M^{-1}$). Finally, $[T]_C^C=Ncdot [T]_B^Bcdot M$.
$endgroup$
– Tito Eliatron
Dec 27 '18 at 18:26
1
1
$begingroup$
just see the image of basis C using given transformation.
$endgroup$
– ASHWINI SANKHE
Dec 28 '18 at 7:20
$begingroup$
just see the image of basis C using given transformation.
$endgroup$
– ASHWINI SANKHE
Dec 28 '18 at 7:20
$begingroup$
Thanks, we still did not learn that formula but were already given it in homework.. Have a good weekend!
$endgroup$
– Tegernako
Dec 28 '18 at 10:38
$begingroup$
Thanks, we still did not learn that formula but were already given it in homework.. Have a good weekend!
$endgroup$
– Tegernako
Dec 28 '18 at 10:38
add a comment |
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1
$begingroup$
Find the matrix $M$ of change of basis from $B$ to $C$, and the matrix $N$ of change of basis from $C$ to $B$ (in fact, $N=M^{-1}$). Finally, $[T]_C^C=Ncdot [T]_B^Bcdot M$.
$endgroup$
– Tito Eliatron
Dec 27 '18 at 18:26
1
$begingroup$
just see the image of basis C using given transformation.
$endgroup$
– ASHWINI SANKHE
Dec 28 '18 at 7:20
$begingroup$
Thanks, we still did not learn that formula but were already given it in homework.. Have a good weekend!
$endgroup$
– Tegernako
Dec 28 '18 at 10:38