Find a matrix R α ∈ R 2 × 2 such that f ( x ) = f R α ( x ) for every x ∈ R 2
I'm trying to solve this task.
Let α∈R be given. Consider the linear map f:R^2→R^2,
(x1 --> ( x1·cos(α)−x2·sin(α)
x2) x1.sin(α)+x2·cos(α)).
a) Find a matrix Rα∈R2×2 such that f(x) =f Rα(x) for everyx∈R^2.
b) Interpret the map f, i.e. state what it does to a given vector x∈R^2.
First I thought, I should use this rule
|A| = ad − bc , but I think, It's already used . should i gave α a number like zero and then I will have a matrix and that's it?
linear-algebra matrices linear-transformations
asked Nov 25 '18 at 22:03
Amerov
35
add a comment |
I'm trying to solve this task.
Let α∈R be given. Consider the linear map f:R^2→R^2,
(x1 --> ( x1·cos(α)−x2·sin(α)
x2) x1.sin(α)+x2·cos(α)).
a) Find a matrix Rα∈R2×2 such that f(x) =f Rα(x) for everyx∈R^2.
b) Interpret the map f, i.e. state what it does to a given vector x∈R^2.
First I thought, I should use this rule
|A| = ad − bc , but I think, It's already used . should i gave α a number like zero and then I will have a matrix and that's it?
linear-algebra matrices linear-transformations
asked Nov 25 '18 at 22:03
Amerov
35
Your question is very difficult to read! You really have to learn MathJax. See this link for a good tutorial.
– TonyK
Nov 25 '18 at 22:08
You have a description there of the result of applying $f$ to an arbitrary column vector $(x_1,x_2)^T$. Can you write that result as a matrix times $(x_1,x_2)^T$ in any way?
– Arthur
Nov 25 '18 at 22:08
@TonyK thanks..
– Amerov
Nov 25 '18 at 22:25
add a comment |
I'm trying to solve this task.
Let α∈R be given. Consider the linear map f:R^2→R^2,
(x1 --> ( x1·cos(α)−x2·sin(α)
x2) x1.sin(α)+x2·cos(α)).
a) Find a matrix Rα∈R2×2 such that f(x) =f Rα(x) for everyx∈R^2.
b) Interpret the map f, i.e. state what it does to a given vector x∈R^2.
First I thought, I should use this rule
|A| = ad − bc , but I think, It's already used . should i gave α a number like zero and then I will have a matrix and that's it?
linear-algebra matrices linear-transformations
asked Nov 25 '18 at 22:03
Amerov
35
I'm trying to solve this task.
Let α∈R be given. Consider the linear map f:R^2→R^2,
(x1 --> ( x1·cos(α)−x2·sin(α)
x2) x1.sin(α)+x2·cos(α)).
a) Find a matrix Rα∈R2×2 such that f(x) =f Rα(x) for everyx∈R^2.
b) Interpret the map f, i.e. state what it does to a given vector x∈R^2.
First I thought, I should use this rule
|A| = ad − bc , but I think, It's already used . should i gave α a number like zero and then I will have a matrix and that's it?
linear-algebra matrices linear-transformations
linear-algebra matrices linear-transformations
asked Nov 25 '18 at 22:03
Amerov
35
asked Nov 25 '18 at 22:03
Amerov
35
edited Nov 25 '18 at 22:10
asked Nov 25 '18 at 22:03
Amerov
35
asked Nov 25 '18 at 22:03
Amerov
35
asked Nov 25 '18 at 22:03
Amerov
35
35
Your question is very difficult to read! You really have to learn MathJax. See this link for a good tutorial.
– TonyK
Nov 25 '18 at 22:08
You have a description there of the result of applying $f$ to an arbitrary column vector $(x_1,x_2)^T$. Can you write that result as a matrix times $(x_1,x_2)^T$ in any way?
– Arthur
Nov 25 '18 at 22:08
@TonyK thanks..
– Amerov
Nov 25 '18 at 22:25
add a comment |
Your question is very difficult to read! You really have to learn MathJax. See this link for a good tutorial.
– TonyK
Nov 25 '18 at 22:08
You have a description there of the result of applying $f$ to an arbitrary column vector $(x_1,x_2)^T$. Can you write that result as a matrix times $(x_1,x_2)^T$ in any way?
– Arthur
Nov 25 '18 at 22:08
@TonyK thanks..
– Amerov
Nov 25 '18 at 22:25
Your question is very difficult to read! You really have to learn MathJax. See this link for a good tutorial.
– TonyK
Nov 25 '18 at 22:08
Your question is very difficult to read! You really have to learn MathJax. See this link for a good tutorial.
– TonyK
Nov 25 '18 at 22:08
You have a description there of the result of applying $f$ to an arbitrary column vector $(x_1,x_2)^T$. Can you write that result as a matrix times $(x_1,x_2)^T$ in any way?
– Arthur
Nov 25 '18 at 22:08
You have a description there of the result of applying $f$ to an arbitrary column vector $(x_1,x_2)^T$. Can you write that result as a matrix times $(x_1,x_2)^T$ in any way?
– Arthur
Nov 25 '18 at 22:08
@TonyK thanks..
– Amerov
Nov 25 '18 at 22:25
@TonyK thanks..
– Amerov
Nov 25 '18 at 22:25
add a comment |
1 Answer
1
active
oldest
votes
Hint: The columns of the matrix are $f(1,0)$ and $f(0,1)$.
For the second part, look at what happens to $(1,0)$ and $(0,1)$ and apply a little trigonometry.
answered Nov 25 '18 at 22:13
Chris Custer
10.8k3824
How did you know that the columns of the m. are f(1,0)and f(0,1)?
– Amerov
Nov 25 '18 at 22:24
This is always true.
– Chris Custer
Nov 25 '18 at 22:27
can you give me a simple example to know how the result should look like ?
– Amerov
Nov 25 '18 at 22:33
For the first column, $f(1,0)=(costheta,sintheta)$. So we have $begin{pmatrix}costheta &*\sintheta &*end{pmatrix}$, so far. You try the second column.
– Chris Custer
Nov 25 '18 at 22:39
it should be (x1,x2) if it's wrong please don't blame me XD XD XD
– Amerov
Nov 25 '18 at 22:46
|
show 23 more comments
Your Answer
StackExchange.ifUsing("editor", function () {
return StackExchange.using("mathjaxEditing", function () {
StackExchange.MarkdownEditor.creationCallbacks.add(function (editor, postfix) {
StackExchange.mathjaxEditing.prepareWmdForMathJax(editor, postfix, [["$", "$"], ["\\(","\\)"]]);
});
});
}, "mathjax-editing");
StackExchange.ready(function() {
var channelOptions = {
tags: "".split(" "),
id: "69"
};
initTagRenderer("".split(" "), "".split(" "), channelOptions);
StackExchange.using("externalEditor", function() {
// Have to fire editor after snippets, if snippets enabled
if (StackExchange.settings.snippets.snippetsEnabled) {
StackExchange.using("snippets", function() {
createEditor();
});
}
else {
createEditor();
}
});
function createEditor() {
StackExchange.prepareEditor({
heartbeatType: 'answer',
autoActivateHeartbeat: false,
convertImagesToLinks: true,
noModals: true,
showLowRepImageUploadWarning: true,
reputationToPostImages: 10,
bindNavPrevention: true,
postfix: "",
imageUploader: {
brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
allowUrls: true
},
noCode: true, onDemand: true,
discardSelector: ".discard-answer"
,immediatelyShowMarkdownHelp:true
});
}
});
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3013476%2ffind-a-matrix-r-%25ce%25b1-%25e2%2588%2588-r-2-%25c3%2597-2-such-that-f-x-f-r-%25ce%25b1-x-for-every-x-%25e2%2588%2588-r-2%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
Hint: The columns of the matrix are $f(1,0)$ and $f(0,1)$.
For the second part, look at what happens to $(1,0)$ and $(0,1)$ and apply a little trigonometry.
answered Nov 25 '18 at 22:13
Chris Custer
10.8k3824
How did you know that the columns of the m. are f(1,0)and f(0,1)?
– Amerov
Nov 25 '18 at 22:24
This is always true.
– Chris Custer
Nov 25 '18 at 22:27
can you give me a simple example to know how the result should look like ?
– Amerov
Nov 25 '18 at 22:33
For the first column, $f(1,0)=(costheta,sintheta)$. So we have $begin{pmatrix}costheta &*\sintheta &*end{pmatrix}$, so far. You try the second column.
– Chris Custer
Nov 25 '18 at 22:39
it should be (x1,x2) if it's wrong please don't blame me XD XD XD
– Amerov
Nov 25 '18 at 22:46
|
show 23 more comments
Hint: The columns of the matrix are $f(1,0)$ and $f(0,1)$.
For the second part, look at what happens to $(1,0)$ and $(0,1)$ and apply a little trigonometry.
answered Nov 25 '18 at 22:13
Chris Custer
10.8k3824
How did you know that the columns of the m. are f(1,0)and f(0,1)?
– Amerov
Nov 25 '18 at 22:24
This is always true.
– Chris Custer
Nov 25 '18 at 22:27
can you give me a simple example to know how the result should look like ?
– Amerov
Nov 25 '18 at 22:33
For the first column, $f(1,0)=(costheta,sintheta)$. So we have $begin{pmatrix}costheta &*\sintheta &*end{pmatrix}$, so far. You try the second column.
– Chris Custer
Nov 25 '18 at 22:39
it should be (x1,x2) if it's wrong please don't blame me XD XD XD
– Amerov
Nov 25 '18 at 22:46
|
show 23 more comments
Hint: The columns of the matrix are $f(1,0)$ and $f(0,1)$.
For the second part, look at what happens to $(1,0)$ and $(0,1)$ and apply a little trigonometry.
answered Nov 25 '18 at 22:13
Chris Custer
10.8k3824
Hint: The columns of the matrix are $f(1,0)$ and $f(0,1)$.
For the second part, look at what happens to $(1,0)$ and $(0,1)$ and apply a little trigonometry.
answered Nov 25 '18 at 22:13
Chris Custer
10.8k3824
answered Nov 25 '18 at 22:13
Chris Custer
10.8k3824
answered Nov 25 '18 at 22:13
Chris Custer
10.8k3824
answered Nov 25 '18 at 22:13
Chris Custer
10.8k3824
10.8k3824
How did you know that the columns of the m. are f(1,0)and f(0,1)?
– Amerov
Nov 25 '18 at 22:24
This is always true.
– Chris Custer
Nov 25 '18 at 22:27
can you give me a simple example to know how the result should look like ?
– Amerov
Nov 25 '18 at 22:33
For the first column, $f(1,0)=(costheta,sintheta)$. So we have $begin{pmatrix}costheta &*\sintheta &*end{pmatrix}$, so far. You try the second column.
– Chris Custer
Nov 25 '18 at 22:39
it should be (x1,x2) if it's wrong please don't blame me XD XD XD
– Amerov
Nov 25 '18 at 22:46
|
show 23 more comments
How did you know that the columns of the m. are f(1,0)and f(0,1)?
– Amerov
Nov 25 '18 at 22:24
This is always true.
– Chris Custer
Nov 25 '18 at 22:27
can you give me a simple example to know how the result should look like ?
– Amerov
Nov 25 '18 at 22:33
For the first column, $f(1,0)=(costheta,sintheta)$. So we have $begin{pmatrix}costheta &*\sintheta &*end{pmatrix}$, so far. You try the second column.
– Chris Custer
Nov 25 '18 at 22:39
it should be (x1,x2) if it's wrong please don't blame me XD XD XD
– Amerov
Nov 25 '18 at 22:46
How did you know that the columns of the m. are f(1,0)and f(0,1)?
– Amerov
Nov 25 '18 at 22:24
How did you know that the columns of the m. are f(1,0)and f(0,1)?
– Amerov
Nov 25 '18 at 22:24
This is always true.
– Chris Custer
Nov 25 '18 at 22:27
This is always true.
– Chris Custer
Nov 25 '18 at 22:27
can you give me a simple example to know how the result should look like ?
– Amerov
Nov 25 '18 at 22:33
can you give me a simple example to know how the result should look like ?
– Amerov
Nov 25 '18 at 22:33
For the first column, $f(1,0)=(costheta,sintheta)$. So we have $begin{pmatrix}costheta &*\sintheta &*end{pmatrix}$, so far. You try the second column.
– Chris Custer
Nov 25 '18 at 22:39
For the first column, $f(1,0)=(costheta,sintheta)$. So we have $begin{pmatrix}costheta &*\sintheta &*end{pmatrix}$, so far. You try the second column.
– Chris Custer
Nov 25 '18 at 22:39
it should be (x1,x2) if it's wrong please don't blame me XD XD XD
– Amerov
Nov 25 '18 at 22:46
it should be (x1,x2) if it's wrong please don't blame me XD XD XD
– Amerov
Nov 25 '18 at 22:46
|
show 23 more comments
Thanks for contributing an answer to Mathematics Stack Exchange!
- Please be sure to answer the question. Provide details and share your research!
But avoid …
- Asking for help, clarification, or responding to other answers.
- Making statements based on opinion; back them up with references or personal experience.
Use MathJax to format equations. MathJax reference.
To learn more, see our tips on writing great answers.
Some of your past answers have not been well-received, and you're in danger of being blocked from answering.
Please pay close attention to the following guidance:
- Please be sure to answer the question. Provide details and share your research!
But avoid …
- Asking for help, clarification, or responding to other answers.
- Making statements based on opinion; back them up with references or personal experience.
To learn more, see our tips on writing great answers.
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3013476%2ffind-a-matrix-r-%25ce%25b1-%25e2%2588%2588-r-2-%25c3%2597-2-such-that-f-x-f-r-%25ce%25b1-x-for-every-x-%25e2%2588%2588-r-2%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Your question is very difficult to read! You really have to learn MathJax. See this link for a good tutorial.
– TonyK
Nov 25 '18 at 22:08
You have a description there of the result of applying $f$ to an arbitrary column vector $(x_1,x_2)^T$. Can you write that result as a matrix times $(x_1,x_2)^T$ in any way?
– Arthur
Nov 25 '18 at 22:08
@TonyK thanks..
– Amerov
Nov 25 '18 at 22:25