Mixing
Find the difference between the number of rows and columns of a matrix.
$begingroup$
Let $B$ be an $n$ by $r$ matrix and let $M$ be the block matrix: begin{pmatrix} 0 &B\ B^t & 0 end{pmatrix}
$M$ has nullity 1. Show that the difference between the number of rows and columns of $B$ is 1.
I am supposed to use the fact that row rank = column rank for matrices over R, however I have no idea where to start.
linear-algebra linear-transformations
asked Jan 4 at 21:17
John CoxJohn Cox
83
$endgroup$
add a comment |
$begingroup$
Let $B$ be an $n$ by $r$ matrix and let $M$ be the block matrix: begin{pmatrix} 0 &B\ B^t & 0 end{pmatrix}
$M$ has nullity 1. Show that the difference between the number of rows and columns of $B$ is 1.
I am supposed to use the fact that row rank = column rank for matrices over R, however I have no idea where to start.
linear-algebra linear-transformations
asked Jan 4 at 21:17
John CoxJohn Cox
83
$endgroup$
2
$begingroup$
How are the nullities of $B$ and $B^t$ related?
$endgroup$
– user3482749
Jan 4 at 21:23
$begingroup$
Since their ranks are equal then Null(B)= Null(B^t)+r-n
$endgroup$
– John Cox
Jan 5 at 10:19
add a comment |
$begingroup$
Let $B$ be an $n$ by $r$ matrix and let $M$ be the block matrix: begin{pmatrix} 0 &B\ B^t & 0 end{pmatrix}
$M$ has nullity 1. Show that the difference between the number of rows and columns of $B$ is 1.
I am supposed to use the fact that row rank = column rank for matrices over R, however I have no idea where to start.
linear-algebra linear-transformations
asked Jan 4 at 21:17
John CoxJohn Cox
83
$endgroup$
Let $B$ be an $n$ by $r$ matrix and let $M$ be the block matrix: begin{pmatrix} 0 &B\ B^t & 0 end{pmatrix}
$M$ has nullity 1. Show that the difference between the number of rows and columns of $B$ is 1.
I am supposed to use the fact that row rank = column rank for matrices over R, however I have no idea where to start.
linear-algebra linear-transformations
linear-algebra linear-transformations
asked Jan 4 at 21:17
John CoxJohn Cox
83
asked Jan 4 at 21:17
John CoxJohn Cox
83
asked Jan 4 at 21:17
John CoxJohn Cox
83
asked Jan 4 at 21:17
John CoxJohn Cox
83
asked Jan 4 at 21:17
John CoxJohn Cox
83
83
2
$begingroup$
How are the nullities of $B$ and $B^t$ related?
$endgroup$
– user3482749
Jan 4 at 21:23
$begingroup$
Since their ranks are equal then Null(B)= Null(B^t)+r-n
$endgroup$
– John Cox
Jan 5 at 10:19
add a comment |
2
$begingroup$
How are the nullities of $B$ and $B^t$ related?
$endgroup$
– user3482749
Jan 4 at 21:23
$begingroup$
Since their ranks are equal then Null(B)= Null(B^t)+r-n
$endgroup$
– John Cox
Jan 5 at 10:19
2
2
$begingroup$
How are the nullities of $B$ and $B^t$ related?
$endgroup$
– user3482749
Jan 4 at 21:23
$begingroup$
How are the nullities of $B$ and $B^t$ related?
$endgroup$
– user3482749
Jan 4 at 21:23
$begingroup$
Since their ranks are equal then Null(B)= Null(B^t)+r-n
$endgroup$
– John Cox
Jan 5 at 10:19
$begingroup$
Since their ranks are equal then Null(B)= Null(B^t)+r-n
$endgroup$
– John Cox
Jan 5 at 10:19
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%2f3062122%2ffind-the-difference-between-the-number-of-rows-and-columns-of-a-matrix%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%2f3062122%2ffind-the-difference-between-the-number-of-rows-and-columns-of-a-matrix%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
2
$begingroup$
How are the nullities of $B$ and $B^t$ related?
$endgroup$
– user3482749
Jan 4 at 21:23
$begingroup$
Since their ranks are equal then Null(B)= Null(B^t)+r-n
$endgroup$
– John Cox
Jan 5 at 10:19