Find a linear transformation defined by $T(0,1,2) = (3,1,2)$ and $T(1,1,1) = (2,2,2)$. [closed]
$begingroup$
The transformation here (as per my calculations) will be $Tcolon U rightarrow V$ such that $T(x,y,z) = (y+z, 3y-z, 2y)$ where $z=-x+2y$.
Now what should $U$ be $Bbb R^3$ or a subset of $Bbb R^3$?
linear-algebra linear-transformations
edited Dec 6 '18 at 5:14
Tianlalu
3,08621038
asked Dec 6 '18 at 4:45
Ajay ChoudharyAjay Choudhary
788
$endgroup$
closed as off-topic by Saad, John B, Brahadeesh, ncmathsadist, GNUSupporter 8964民主女神 地下教會 Dec 10 '18 at 14:07
This question appears to be off-topic. The users who voted to close gave this specific reason:
- "This question is missing context or other details: Please provide additional context, which ideally explains why the question is relevant to you and our community. Some forms of context include: background and motivation, relevant definitions, source, possible strategies, your current progress, why the question is interesting or important, etc." – Saad, John B, Brahadeesh
If this question can be reworded to fit the rules in the help center, please edit the question.
|
show 1 more comment
$begingroup$
The transformation here (as per my calculations) will be $Tcolon U rightarrow V$ such that $T(x,y,z) = (y+z, 3y-z, 2y)$ where $z=-x+2y$.
Now what should $U$ be $Bbb R^3$ or a subset of $Bbb R^3$?
linear-algebra linear-transformations
edited Dec 6 '18 at 5:14
Tianlalu
3,08621038
asked Dec 6 '18 at 4:45
Ajay ChoudharyAjay Choudhary
788
$endgroup$
closed as off-topic by Saad, John B, Brahadeesh, ncmathsadist, GNUSupporter 8964民主女神 地下教會 Dec 10 '18 at 14:07
This question appears to be off-topic. The users who voted to close gave this specific reason:
- "This question is missing context or other details: Please provide additional context, which ideally explains why the question is relevant to you and our community. Some forms of context include: background and motivation, relevant definitions, source, possible strategies, your current progress, why the question is interesting or important, etc." – Saad, John B, Brahadeesh
If this question can be reworded to fit the rules in the help center, please edit the question.
$begingroup$
There will be many linear maps that do this, since you have only specified where a 2 dimensional subspace goes
$endgroup$
– qbert
Dec 6 '18 at 4:57
$begingroup$
@qbert I could only find a single transformation. Can you hint how to find others?
$endgroup$
– Ajay Choudhary
Dec 6 '18 at 5:02
$begingroup$
To find others just extend ${(0,1,2),(1,1,1)}$ to a basis for $mathbb R^3$. For instance, you could use $(1,0,0)$. Then just map it to different vectors to get other examples.
$endgroup$
– Chris Custer
Dec 6 '18 at 5:58
$begingroup$
If this is exactly the problem as it was presented to you (where did it come from, by the way?), then it’s common to assume that $T:mathbb R^3tomathbb R^3$ was intended. If this is from some textbook, compare this exercise to other examples and exercises in the book to be sure.
$endgroup$
– amd
Dec 6 '18 at 6:36
1
$begingroup$
There’s no way for anyone else to know for sure unless you provide a source or some other context for comparison. At any rate, the right person to ask is whomever is going to be grading your work.
$endgroup$
– amd
Dec 6 '18 at 19:15
|
show 1 more comment
$begingroup$
The transformation here (as per my calculations) will be $Tcolon U rightarrow V$ such that $T(x,y,z) = (y+z, 3y-z, 2y)$ where $z=-x+2y$.
Now what should $U$ be $Bbb R^3$ or a subset of $Bbb R^3$?
linear-algebra linear-transformations
edited Dec 6 '18 at 5:14
Tianlalu
3,08621038
asked Dec 6 '18 at 4:45
Ajay ChoudharyAjay Choudhary
788
$endgroup$
The transformation here (as per my calculations) will be $Tcolon U rightarrow V$ such that $T(x,y,z) = (y+z, 3y-z, 2y)$ where $z=-x+2y$.
Now what should $U$ be $Bbb R^3$ or a subset of $Bbb R^3$?
linear-algebra linear-transformations
linear-algebra linear-transformations
edited Dec 6 '18 at 5:14
Tianlalu
3,08621038
asked Dec 6 '18 at 4:45
Ajay ChoudharyAjay Choudhary
788
edited Dec 6 '18 at 5:14
Tianlalu
3,08621038
asked Dec 6 '18 at 4:45
Ajay ChoudharyAjay Choudhary
788
edited Dec 6 '18 at 5:14
Tianlalu
3,08621038
edited Dec 6 '18 at 5:14
Tianlalu
3,08621038
edited Dec 6 '18 at 5:14
Tianlalu
3,08621038
3,08621038
asked Dec 6 '18 at 4:45
Ajay ChoudharyAjay Choudhary
788
asked Dec 6 '18 at 4:45
Ajay ChoudharyAjay Choudhary
788
asked Dec 6 '18 at 4:45
Ajay ChoudharyAjay Choudhary
788
788
closed as off-topic by Saad, John B, Brahadeesh, ncmathsadist, GNUSupporter 8964民主女神 地下教會 Dec 10 '18 at 14:07
This question appears to be off-topic. The users who voted to close gave this specific reason:
- "This question is missing context or other details: Please provide additional context, which ideally explains why the question is relevant to you and our community. Some forms of context include: background and motivation, relevant definitions, source, possible strategies, your current progress, why the question is interesting or important, etc." – Saad, John B, Brahadeesh
If this question can be reworded to fit the rules in the help center, please edit the question.
closed as off-topic by Saad, John B, Brahadeesh, ncmathsadist, GNUSupporter 8964民主女神 地下教會 Dec 10 '18 at 14:07
This question appears to be off-topic. The users who voted to close gave this specific reason:
- "This question is missing context or other details: Please provide additional context, which ideally explains why the question is relevant to you and our community. Some forms of context include: background and motivation, relevant definitions, source, possible strategies, your current progress, why the question is interesting or important, etc." – Saad, John B, Brahadeesh
If this question can be reworded to fit the rules in the help center, please edit the question.
$begingroup$
There will be many linear maps that do this, since you have only specified where a 2 dimensional subspace goes
$endgroup$
– qbert
Dec 6 '18 at 4:57
$begingroup$
@qbert I could only find a single transformation. Can you hint how to find others?
$endgroup$
– Ajay Choudhary
Dec 6 '18 at 5:02
$begingroup$
To find others just extend ${(0,1,2),(1,1,1)}$ to a basis for $mathbb R^3$. For instance, you could use $(1,0,0)$. Then just map it to different vectors to get other examples.
$endgroup$
– Chris Custer
Dec 6 '18 at 5:58
$begingroup$
If this is exactly the problem as it was presented to you (where did it come from, by the way?), then it’s common to assume that $T:mathbb R^3tomathbb R^3$ was intended. If this is from some textbook, compare this exercise to other examples and exercises in the book to be sure.
$endgroup$
– amd
Dec 6 '18 at 6:36
1
$begingroup$
There’s no way for anyone else to know for sure unless you provide a source or some other context for comparison. At any rate, the right person to ask is whomever is going to be grading your work.
$endgroup$
– amd
Dec 6 '18 at 19:15
|
show 1 more comment
$begingroup$
There will be many linear maps that do this, since you have only specified where a 2 dimensional subspace goes
$endgroup$
– qbert
Dec 6 '18 at 4:57
$begingroup$
@qbert I could only find a single transformation. Can you hint how to find others?
$endgroup$
– Ajay Choudhary
Dec 6 '18 at 5:02
$begingroup$
To find others just extend ${(0,1,2),(1,1,1)}$ to a basis for $mathbb R^3$. For instance, you could use $(1,0,0)$. Then just map it to different vectors to get other examples.
$endgroup$
– Chris Custer
Dec 6 '18 at 5:58
$begingroup$
If this is exactly the problem as it was presented to you (where did it come from, by the way?), then it’s common to assume that $T:mathbb R^3tomathbb R^3$ was intended. If this is from some textbook, compare this exercise to other examples and exercises in the book to be sure.
$endgroup$
– amd
Dec 6 '18 at 6:36
1
$begingroup$
There’s no way for anyone else to know for sure unless you provide a source or some other context for comparison. At any rate, the right person to ask is whomever is going to be grading your work.
$endgroup$
– amd
Dec 6 '18 at 19:15
$begingroup$
There will be many linear maps that do this, since you have only specified where a 2 dimensional subspace goes
$endgroup$
– qbert
Dec 6 '18 at 4:57
$begingroup$
There will be many linear maps that do this, since you have only specified where a 2 dimensional subspace goes
$endgroup$
– qbert
Dec 6 '18 at 4:57
$begingroup$
@qbert I could only find a single transformation. Can you hint how to find others?
$endgroup$
– Ajay Choudhary
Dec 6 '18 at 5:02
$begingroup$
@qbert I could only find a single transformation. Can you hint how to find others?
$endgroup$
– Ajay Choudhary
Dec 6 '18 at 5:02
$begingroup$
To find others just extend ${(0,1,2),(1,1,1)}$ to a basis for $mathbb R^3$. For instance, you could use $(1,0,0)$. Then just map it to different vectors to get other examples.
$endgroup$
– Chris Custer
Dec 6 '18 at 5:58
$begingroup$
To find others just extend ${(0,1,2),(1,1,1)}$ to a basis for $mathbb R^3$. For instance, you could use $(1,0,0)$. Then just map it to different vectors to get other examples.
$endgroup$
– Chris Custer
Dec 6 '18 at 5:58
$begingroup$
If this is exactly the problem as it was presented to you (where did it come from, by the way?), then it’s common to assume that $T:mathbb R^3tomathbb R^3$ was intended. If this is from some textbook, compare this exercise to other examples and exercises in the book to be sure.
$endgroup$
– amd
Dec 6 '18 at 6:36
$begingroup$
If this is exactly the problem as it was presented to you (where did it come from, by the way?), then it’s common to assume that $T:mathbb R^3tomathbb R^3$ was intended. If this is from some textbook, compare this exercise to other examples and exercises in the book to be sure.
$endgroup$
– amd
Dec 6 '18 at 6:36
1
1
$begingroup$
There’s no way for anyone else to know for sure unless you provide a source or some other context for comparison. At any rate, the right person to ask is whomever is going to be grading your work.
$endgroup$
– amd
Dec 6 '18 at 19:15
$begingroup$
There’s no way for anyone else to know for sure unless you provide a source or some other context for comparison. At any rate, the right person to ask is whomever is going to be grading your work.
$endgroup$
– amd
Dec 6 '18 at 19:15
|
show 1 more comment
1 Answer
1
active
oldest
votes
$begingroup$
If you want a linear transformation $T$ such that $T(u_1)=v_1$ and $T(u_2)=v_2$ where $u_1$ and $u_2$ are linearly independent, then we want $T(alpha u_1+beta u_2)=alpha T(u_1)+beta T(u_2)$ for scalars $alpha,beta$, and then $T$ is defined on the subspace $operatorname{span}(u_1,u_2)$. Also, the range will be $operatorname{span}(v_1,v_2)$.
answered Dec 6 '18 at 4:55
DaveDave
8,76711033
$endgroup$
$begingroup$
Yes, I was also thinking the same, but needed some corroboration. Thanks.
$endgroup$
– Ajay Choudhary
Dec 6 '18 at 5:00
add a comment |
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
If you want a linear transformation $T$ such that $T(u_1)=v_1$ and $T(u_2)=v_2$ where $u_1$ and $u_2$ are linearly independent, then we want $T(alpha u_1+beta u_2)=alpha T(u_1)+beta T(u_2)$ for scalars $alpha,beta$, and then $T$ is defined on the subspace $operatorname{span}(u_1,u_2)$. Also, the range will be $operatorname{span}(v_1,v_2)$.
answered Dec 6 '18 at 4:55
DaveDave
8,76711033
$endgroup$
$begingroup$
Yes, I was also thinking the same, but needed some corroboration. Thanks.
$endgroup$
– Ajay Choudhary
Dec 6 '18 at 5:00
add a comment |
$begingroup$
If you want a linear transformation $T$ such that $T(u_1)=v_1$ and $T(u_2)=v_2$ where $u_1$ and $u_2$ are linearly independent, then we want $T(alpha u_1+beta u_2)=alpha T(u_1)+beta T(u_2)$ for scalars $alpha,beta$, and then $T$ is defined on the subspace $operatorname{span}(u_1,u_2)$. Also, the range will be $operatorname{span}(v_1,v_2)$.
answered Dec 6 '18 at 4:55
DaveDave
8,76711033
$endgroup$
$begingroup$
Yes, I was also thinking the same, but needed some corroboration. Thanks.
$endgroup$
– Ajay Choudhary
Dec 6 '18 at 5:00
add a comment |
$begingroup$
If you want a linear transformation $T$ such that $T(u_1)=v_1$ and $T(u_2)=v_2$ where $u_1$ and $u_2$ are linearly independent, then we want $T(alpha u_1+beta u_2)=alpha T(u_1)+beta T(u_2)$ for scalars $alpha,beta$, and then $T$ is defined on the subspace $operatorname{span}(u_1,u_2)$. Also, the range will be $operatorname{span}(v_1,v_2)$.
answered Dec 6 '18 at 4:55
DaveDave
8,76711033
$endgroup$
If you want a linear transformation $T$ such that $T(u_1)=v_1$ and $T(u_2)=v_2$ where $u_1$ and $u_2$ are linearly independent, then we want $T(alpha u_1+beta u_2)=alpha T(u_1)+beta T(u_2)$ for scalars $alpha,beta$, and then $T$ is defined on the subspace $operatorname{span}(u_1,u_2)$. Also, the range will be $operatorname{span}(v_1,v_2)$.
answered Dec 6 '18 at 4:55
DaveDave
8,76711033
answered Dec 6 '18 at 4:55
DaveDave
8,76711033
answered Dec 6 '18 at 4:55
DaveDave
8,76711033
answered Dec 6 '18 at 4:55
DaveDave
8,76711033
8,76711033
$begingroup$
Yes, I was also thinking the same, but needed some corroboration. Thanks.
$endgroup$
– Ajay Choudhary
Dec 6 '18 at 5:00
add a comment |
$begingroup$
Yes, I was also thinking the same, but needed some corroboration. Thanks.
$endgroup$
– Ajay Choudhary
Dec 6 '18 at 5:00
$begingroup$
Yes, I was also thinking the same, but needed some corroboration. Thanks.
$endgroup$
– Ajay Choudhary
Dec 6 '18 at 5:00
$begingroup$
Yes, I was also thinking the same, but needed some corroboration. Thanks.
$endgroup$
– Ajay Choudhary
Dec 6 '18 at 5:00
add a comment |
$begingroup$
There will be many linear maps that do this, since you have only specified where a 2 dimensional subspace goes
$endgroup$
– qbert
Dec 6 '18 at 4:57
$begingroup$
@qbert I could only find a single transformation. Can you hint how to find others?
$endgroup$
– Ajay Choudhary
Dec 6 '18 at 5:02
$begingroup$
To find others just extend ${(0,1,2),(1,1,1)}$ to a basis for $mathbb R^3$. For instance, you could use $(1,0,0)$. Then just map it to different vectors to get other examples.
$endgroup$
– Chris Custer
Dec 6 '18 at 5:58
$begingroup$
If this is exactly the problem as it was presented to you (where did it come from, by the way?), then it’s common to assume that $T:mathbb R^3tomathbb R^3$ was intended. If this is from some textbook, compare this exercise to other examples and exercises in the book to be sure.
$endgroup$
– amd
Dec 6 '18 at 6:36
1
$begingroup$
There’s no way for anyone else to know for sure unless you provide a source or some other context for comparison. At any rate, the right person to ask is whomever is going to be grading your work.
$endgroup$
– amd
Dec 6 '18 at 19:15