Show that $operatorname{rk}(g circ f) le min(operatorname{rk}(f), operatorname{rk}(g))$
$begingroup$
Show that for finite K-vector spaces U, V, W with $f:U rightarrow V$ and $g:V rightarrow W$ linear it follows that:
$$operatorname{rank}(g circ f) le min(operatorname{rank}(f), operatorname{rank}(g))$$
Hm ok so we know that:
$$operatorname{rank}(f)=dim_K(operatorname{im}(f))$$
$$operatorname{rank}(g)=dim_K(operatorname{im}(g))$$
$$operatorname{rank}(g circ f)=dim_K(operatorname{im}(g circ f))$$
I don't really know how to proceed here. Can somebody give me a hint?
linear-algebra proof-writing
edited Dec 14 '18 at 18:22
Carl Schildkraut
11.2k11441
asked Dec 14 '18 at 17:57
D. JohnD. John
283
$endgroup$
add a comment |
$begingroup$
Show that for finite K-vector spaces U, V, W with $f:U rightarrow V$ and $g:V rightarrow W$ linear it follows that:
$$operatorname{rank}(g circ f) le min(operatorname{rank}(f), operatorname{rank}(g))$$
Hm ok so we know that:
$$operatorname{rank}(f)=dim_K(operatorname{im}(f))$$
$$operatorname{rank}(g)=dim_K(operatorname{im}(g))$$
$$operatorname{rank}(g circ f)=dim_K(operatorname{im}(g circ f))$$
I don't really know how to proceed here. Can somebody give me a hint?
linear-algebra proof-writing
edited Dec 14 '18 at 18:22
Carl Schildkraut
11.2k11441
asked Dec 14 '18 at 17:57
D. JohnD. John
283
$endgroup$
$begingroup$
What do you mean by $rg$? Rank?
$endgroup$
– Naweed G. Seldon
Dec 14 '18 at 18:02
$begingroup$
Yes. Sorry I didn't know this isn't standard notation.
$endgroup$
– D. John
Dec 14 '18 at 18:05
add a comment |
$begingroup$
Show that for finite K-vector spaces U, V, W with $f:U rightarrow V$ and $g:V rightarrow W$ linear it follows that:
$$operatorname{rank}(g circ f) le min(operatorname{rank}(f), operatorname{rank}(g))$$
Hm ok so we know that:
$$operatorname{rank}(f)=dim_K(operatorname{im}(f))$$
$$operatorname{rank}(g)=dim_K(operatorname{im}(g))$$
$$operatorname{rank}(g circ f)=dim_K(operatorname{im}(g circ f))$$
I don't really know how to proceed here. Can somebody give me a hint?
linear-algebra proof-writing
edited Dec 14 '18 at 18:22
Carl Schildkraut
11.2k11441
asked Dec 14 '18 at 17:57
D. JohnD. John
283
$endgroup$
Show that for finite K-vector spaces U, V, W with $f:U rightarrow V$ and $g:V rightarrow W$ linear it follows that:
$$operatorname{rank}(g circ f) le min(operatorname{rank}(f), operatorname{rank}(g))$$
Hm ok so we know that:
$$operatorname{rank}(f)=dim_K(operatorname{im}(f))$$
$$operatorname{rank}(g)=dim_K(operatorname{im}(g))$$
$$operatorname{rank}(g circ f)=dim_K(operatorname{im}(g circ f))$$
I don't really know how to proceed here. Can somebody give me a hint?
linear-algebra proof-writing
linear-algebra proof-writing
edited Dec 14 '18 at 18:22
Carl Schildkraut
11.2k11441
asked Dec 14 '18 at 17:57
D. JohnD. John
283
edited Dec 14 '18 at 18:22
Carl Schildkraut
11.2k11441
asked Dec 14 '18 at 17:57
D. JohnD. John
283
edited Dec 14 '18 at 18:22
Carl Schildkraut
11.2k11441
edited Dec 14 '18 at 18:22
Carl Schildkraut
11.2k11441
edited Dec 14 '18 at 18:22
Carl Schildkraut
11.2k11441
11.2k11441
asked Dec 14 '18 at 17:57
D. JohnD. John
283
asked Dec 14 '18 at 17:57
D. JohnD. John
283
asked Dec 14 '18 at 17:57
D. JohnD. John
283
283
$begingroup$
What do you mean by $rg$? Rank?
$endgroup$
– Naweed G. Seldon
Dec 14 '18 at 18:02
$begingroup$
Yes. Sorry I didn't know this isn't standard notation.
$endgroup$
– D. John
Dec 14 '18 at 18:05
add a comment |
$begingroup$
What do you mean by $rg$? Rank?
$endgroup$
– Naweed G. Seldon
Dec 14 '18 at 18:02
$begingroup$
Yes. Sorry I didn't know this isn't standard notation.
$endgroup$
– D. John
Dec 14 '18 at 18:05
$begingroup$
What do you mean by $rg$? Rank?
$endgroup$
– Naweed G. Seldon
Dec 14 '18 at 18:02
$begingroup$
What do you mean by $rg$? Rank?
$endgroup$
– Naweed G. Seldon
Dec 14 '18 at 18:02
$begingroup$
Yes. Sorry I didn't know this isn't standard notation.
$endgroup$
– D. John
Dec 14 '18 at 18:05
$begingroup$
Yes. Sorry I didn't know this isn't standard notation.
$endgroup$
– D. John
Dec 14 '18 at 18:05
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
Here's a hint: if $f : V to W$ is a linear transformation with a finite rank $r$, then given any list of vectors $v_1, ldots, v_n in V$, the images $f(v_1), ldots, f(v_n)$ can only contain at most $r$ linearly independent vectors. It doesn't matter if $v_1, ldots, v_n$ are linearly independent or not, this still holds true.
Prove this, and examine what happens to a basis of $U$ when mapped first under $f$, then under $g$.
answered Dec 14 '18 at 18:17
Theo BenditTheo Bendit
17.2k12149
$endgroup$
add a comment |
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%2f3039711%2fshow-that-operatornamerkg-circ-f-le-min-operatornamerkf-operator%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
$begingroup$
Here's a hint: if $f : V to W$ is a linear transformation with a finite rank $r$, then given any list of vectors $v_1, ldots, v_n in V$, the images $f(v_1), ldots, f(v_n)$ can only contain at most $r$ linearly independent vectors. It doesn't matter if $v_1, ldots, v_n$ are linearly independent or not, this still holds true.
Prove this, and examine what happens to a basis of $U$ when mapped first under $f$, then under $g$.
answered Dec 14 '18 at 18:17
Theo BenditTheo Bendit
17.2k12149
$endgroup$
add a comment |
$begingroup$
Here's a hint: if $f : V to W$ is a linear transformation with a finite rank $r$, then given any list of vectors $v_1, ldots, v_n in V$, the images $f(v_1), ldots, f(v_n)$ can only contain at most $r$ linearly independent vectors. It doesn't matter if $v_1, ldots, v_n$ are linearly independent or not, this still holds true.
Prove this, and examine what happens to a basis of $U$ when mapped first under $f$, then under $g$.
answered Dec 14 '18 at 18:17
Theo BenditTheo Bendit
17.2k12149
$endgroup$
add a comment |
$begingroup$
Here's a hint: if $f : V to W$ is a linear transformation with a finite rank $r$, then given any list of vectors $v_1, ldots, v_n in V$, the images $f(v_1), ldots, f(v_n)$ can only contain at most $r$ linearly independent vectors. It doesn't matter if $v_1, ldots, v_n$ are linearly independent or not, this still holds true.
Prove this, and examine what happens to a basis of $U$ when mapped first under $f$, then under $g$.
answered Dec 14 '18 at 18:17
Theo BenditTheo Bendit
17.2k12149
$endgroup$
Here's a hint: if $f : V to W$ is a linear transformation with a finite rank $r$, then given any list of vectors $v_1, ldots, v_n in V$, the images $f(v_1), ldots, f(v_n)$ can only contain at most $r$ linearly independent vectors. It doesn't matter if $v_1, ldots, v_n$ are linearly independent or not, this still holds true.
Prove this, and examine what happens to a basis of $U$ when mapped first under $f$, then under $g$.
answered Dec 14 '18 at 18:17
Theo BenditTheo Bendit
17.2k12149
answered Dec 14 '18 at 18:17
Theo BenditTheo Bendit
17.2k12149
answered Dec 14 '18 at 18:17
Theo BenditTheo Bendit
17.2k12149
answered Dec 14 '18 at 18:17
Theo BenditTheo Bendit
17.2k12149
17.2k12149
add a comment |
add a comment |
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%2f3039711%2fshow-that-operatornamerkg-circ-f-le-min-operatornamerkf-operator%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
$begingroup$
What do you mean by $rg$? Rank?
$endgroup$
– Naweed G. Seldon
Dec 14 '18 at 18:02
$begingroup$
Yes. Sorry I didn't know this isn't standard notation.
$endgroup$
– D. John
Dec 14 '18 at 18:05