Isomorphism and dimension exercise clarification.
$begingroup$
I'd need a clarification on one of the statements here below.
Given that V is the set of linear operators f: $mathbb{R}^k rightarrow mathbb{R}^n$ , and I'm trying to prove that V has same dimension as $mathbb{R} ^{kn}$ so that to prove it.
Now, why is $ m=kn$ ? And also why is V contained in $(mathbb{R}^{n})^{(mathbb{R}^k)} ?$
I'm really looking for the reasoning (the simpler the better) behind these statements.
linear-algebra linear-transformations dimension-theory vector-space-isomorphism
asked Dec 7 '18 at 11:53
RicouelloRicouello
1355
$endgroup$
add a comment |
$begingroup$
I'd need a clarification on one of the statements here below.
Given that V is the set of linear operators f: $mathbb{R}^k rightarrow mathbb{R}^n$ , and I'm trying to prove that V has same dimension as $mathbb{R} ^{kn}$ so that to prove it.
Now, why is $ m=kn$ ? And also why is V contained in $(mathbb{R}^{n})^{(mathbb{R}^k)} ?$
I'm really looking for the reasoning (the simpler the better) behind these statements.
linear-algebra linear-transformations dimension-theory vector-space-isomorphism
asked Dec 7 '18 at 11:53
RicouelloRicouello
1355
$endgroup$
$begingroup$
that argument is wrong! you need $f_i(A)subset B$, otherwise you are only modelling surjective morphisms.
$endgroup$
– Enkidu
Dec 7 '18 at 12:03
add a comment |
$begingroup$
I'd need a clarification on one of the statements here below.
Given that V is the set of linear operators f: $mathbb{R}^k rightarrow mathbb{R}^n$ , and I'm trying to prove that V has same dimension as $mathbb{R} ^{kn}$ so that to prove it.
Now, why is $ m=kn$ ? And also why is V contained in $(mathbb{R}^{n})^{(mathbb{R}^k)} ?$
I'm really looking for the reasoning (the simpler the better) behind these statements.
linear-algebra linear-transformations dimension-theory vector-space-isomorphism
asked Dec 7 '18 at 11:53
RicouelloRicouello
1355
$endgroup$
I'd need a clarification on one of the statements here below.
Given that V is the set of linear operators f: $mathbb{R}^k rightarrow mathbb{R}^n$ , and I'm trying to prove that V has same dimension as $mathbb{R} ^{kn}$ so that to prove it.
Now, why is $ m=kn$ ? And also why is V contained in $(mathbb{R}^{n})^{(mathbb{R}^k)} ?$
I'm really looking for the reasoning (the simpler the better) behind these statements.
linear-algebra linear-transformations dimension-theory vector-space-isomorphism
linear-algebra linear-transformations dimension-theory vector-space-isomorphism
asked Dec 7 '18 at 11:53
RicouelloRicouello
1355
asked Dec 7 '18 at 11:53
RicouelloRicouello
1355
asked Dec 7 '18 at 11:53
RicouelloRicouello
1355
asked Dec 7 '18 at 11:53
RicouelloRicouello
1355
asked Dec 7 '18 at 11:53
RicouelloRicouello
1355
1355
$begingroup$
that argument is wrong! you need $f_i(A)subset B$, otherwise you are only modelling surjective morphisms.
$endgroup$
– Enkidu
Dec 7 '18 at 12:03
add a comment |
$begingroup$
that argument is wrong! you need $f_i(A)subset B$, otherwise you are only modelling surjective morphisms.
$endgroup$
– Enkidu
Dec 7 '18 at 12:03
$begingroup$
that argument is wrong! you need $f_i(A)subset B$, otherwise you are only modelling surjective morphisms.
$endgroup$
– Enkidu
Dec 7 '18 at 12:03
$begingroup$
that argument is wrong! you need $f_i(A)subset B$, otherwise you are only modelling surjective morphisms.
$endgroup$
– Enkidu
Dec 7 '18 at 12:03
add a comment |
0
active
oldest
votes
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%2f3029820%2fisomorphism-and-dimension-exercise-clarification%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
0
active
oldest
votes
0
active
oldest
votes
active
oldest
votes
active
oldest
votes
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.
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%2f3029820%2fisomorphism-and-dimension-exercise-clarification%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
$begingroup$
that argument is wrong! you need $f_i(A)subset B$, otherwise you are only modelling surjective morphisms.
$endgroup$
– Enkidu
Dec 7 '18 at 12:03