Eigenspace and dimensions of linear transformation in complex plane
$begingroup$
Suppose V is a finite-dimensional complex vector space and T : V → V is a linear transformation which satisfies $T^{2} = T$. Prove that for any vector v $in$ V , the vector T(v) is contained in the eigenspace $E_{1}$ of eigenvalue $1$ and v −T(v) is contained in the eigenspace $E_{0}$ of eigenvalue $0$. Also prove that T is diagonalizable, and compute the dimensions of $E_{0}$ and $E_{1}$ in terms of the rank of T.
From $T^{2}-T=0$, I could get $lambda=0$ or $lambda= 1$ but I do not know how to continue.
linear-algebra linear-transformations
asked Dec 10 '18 at 13:56
AdamAdam
1
$endgroup$
add a comment |
$begingroup$
Suppose V is a finite-dimensional complex vector space and T : V → V is a linear transformation which satisfies $T^{2} = T$. Prove that for any vector v $in$ V , the vector T(v) is contained in the eigenspace $E_{1}$ of eigenvalue $1$ and v −T(v) is contained in the eigenspace $E_{0}$ of eigenvalue $0$. Also prove that T is diagonalizable, and compute the dimensions of $E_{0}$ and $E_{1}$ in terms of the rank of T.
From $T^{2}-T=0$, I could get $lambda=0$ or $lambda= 1$ but I do not know how to continue.
linear-algebra linear-transformations
asked Dec 10 '18 at 13:56
AdamAdam
1
$endgroup$
add a comment |
$begingroup$
Suppose V is a finite-dimensional complex vector space and T : V → V is a linear transformation which satisfies $T^{2} = T$. Prove that for any vector v $in$ V , the vector T(v) is contained in the eigenspace $E_{1}$ of eigenvalue $1$ and v −T(v) is contained in the eigenspace $E_{0}$ of eigenvalue $0$. Also prove that T is diagonalizable, and compute the dimensions of $E_{0}$ and $E_{1}$ in terms of the rank of T.
From $T^{2}-T=0$, I could get $lambda=0$ or $lambda= 1$ but I do not know how to continue.
linear-algebra linear-transformations
asked Dec 10 '18 at 13:56
AdamAdam
1
$endgroup$
Suppose V is a finite-dimensional complex vector space and T : V → V is a linear transformation which satisfies $T^{2} = T$. Prove that for any vector v $in$ V , the vector T(v) is contained in the eigenspace $E_{1}$ of eigenvalue $1$ and v −T(v) is contained in the eigenspace $E_{0}$ of eigenvalue $0$. Also prove that T is diagonalizable, and compute the dimensions of $E_{0}$ and $E_{1}$ in terms of the rank of T.
From $T^{2}-T=0$, I could get $lambda=0$ or $lambda= 1$ but I do not know how to continue.
linear-algebra linear-transformations
linear-algebra linear-transformations
asked Dec 10 '18 at 13:56
AdamAdam
1
asked Dec 10 '18 at 13:56
AdamAdam
1
asked Dec 10 '18 at 13:56
AdamAdam
1
asked Dec 10 '18 at 13:56
AdamAdam
1
asked Dec 10 '18 at 13:56
AdamAdam
1
1
add a comment |
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
Hint: If $vin V$, then $v=bigl(v-T(v)bigr)+T(v)$.
answered Dec 10 '18 at 14:15
José Carlos SantosJosé Carlos Santos
162k22130233
$endgroup$
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%2f3033942%2feigenspace-and-dimensions-of-linear-transformation-in-complex-plane%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
$begingroup$
Hint: If $vin V$, then $v=bigl(v-T(v)bigr)+T(v)$.
answered Dec 10 '18 at 14:15
José Carlos SantosJosé Carlos Santos
162k22130233
$endgroup$
add a comment |
$begingroup$
Hint: If $vin V$, then $v=bigl(v-T(v)bigr)+T(v)$.
answered Dec 10 '18 at 14:15
José Carlos SantosJosé Carlos Santos
162k22130233
$endgroup$
add a comment |
$begingroup$
Hint: If $vin V$, then $v=bigl(v-T(v)bigr)+T(v)$.
answered Dec 10 '18 at 14:15
José Carlos SantosJosé Carlos Santos
162k22130233
$endgroup$
Hint: If $vin V$, then $v=bigl(v-T(v)bigr)+T(v)$.
answered Dec 10 '18 at 14:15
José Carlos SantosJosé Carlos Santos
162k22130233
answered Dec 10 '18 at 14:15
José Carlos SantosJosé Carlos Santos
162k22130233
answered Dec 10 '18 at 14:15
José Carlos SantosJosé Carlos Santos
162k22130233
answered Dec 10 '18 at 14:15
José Carlos SantosJosé Carlos Santos
162k22130233
162k22130233
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%2f3033942%2feigenspace-and-dimensions-of-linear-transformation-in-complex-plane%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
