Confused with Linear Algebra Proofs












1














I know that this is true, but I am not sure how to prove it:



For $l×m$ matrix $A$ and $m×n$ matrix $B$, it will always be the case that $operatorname{null}(B)$ is a subset of $operatorname{null}(AB)$



My other question is:



Suppose $B$ is a square matrix of size $n$. What are two statements equivalent to $operatorname{nullity}(B) > 0$ ?



Any help is appreciated, thank you!






linear-algebra







share|cite|improve this question
























  • Rank-nullity theorem does it for you.
    – Sean Roberson
    Nov 28 '18 at 3:21










  • The first statement you prove just by checking it..
    – Randall
    Nov 28 '18 at 3:22
















1














I know that this is true, but I am not sure how to prove it:



For $l×m$ matrix $A$ and $m×n$ matrix $B$, it will always be the case that $operatorname{null}(B)$ is a subset of $operatorname{null}(AB)$



My other question is:



Suppose $B$ is a square matrix of size $n$. What are two statements equivalent to $operatorname{nullity}(B) > 0$ ?



Any help is appreciated, thank you!






linear-algebra







share|cite|improve this question
























  • Rank-nullity theorem does it for you.
    – Sean Roberson
    Nov 28 '18 at 3:21










  • The first statement you prove just by checking it..
    – Randall
    Nov 28 '18 at 3:22














1












1








1







I know that this is true, but I am not sure how to prove it:



For $l×m$ matrix $A$ and $m×n$ matrix $B$, it will always be the case that $operatorname{null}(B)$ is a subset of $operatorname{null}(AB)$



My other question is:



Suppose $B$ is a square matrix of size $n$. What are two statements equivalent to $operatorname{nullity}(B) > 0$ ?



Any help is appreciated, thank you!






linear-algebra







share|cite|improve this question















I know that this is true, but I am not sure how to prove it:



For $l×m$ matrix $A$ and $m×n$ matrix $B$, it will always be the case that $operatorname{null}(B)$ is a subset of $operatorname{null}(AB)$



My other question is:



Suppose $B$ is a square matrix of size $n$. What are two statements equivalent to $operatorname{nullity}(B) > 0$ ?



Any help is appreciated, thank you!






linear-algebra




linear-algebra






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited Nov 28 '18 at 4:16









Thomas Shelby

1,830217




1,830217










asked Nov 28 '18 at 3:18









Thomas

62




62












  • Rank-nullity theorem does it for you.
    – Sean Roberson
    Nov 28 '18 at 3:21










  • The first statement you prove just by checking it..
    – Randall
    Nov 28 '18 at 3:22


















  • Rank-nullity theorem does it for you.
    – Sean Roberson
    Nov 28 '18 at 3:21










  • The first statement you prove just by checking it..
    – Randall
    Nov 28 '18 at 3:22
















Rank-nullity theorem does it for you.
– Sean Roberson
Nov 28 '18 at 3:21




Rank-nullity theorem does it for you.
– Sean Roberson
Nov 28 '18 at 3:21












The first statement you prove just by checking it..
– Randall
Nov 28 '18 at 3:22




The first statement you prove just by checking it..
– Randall
Nov 28 '18 at 3:22










1 Answer
1






active

oldest

votes


















1














First, the dimensions of the matrices are irrelevant except for the fact that we want $AB$ to be a well defined matrix product. So you can safely ignore that for the moment.



You want to prove that $null(B)subset null(AB)$. So pick a vector in $null(B)$, say $vec{v}$. What can you say about the matrix-vector product $(AB)vec{v}$?



You also want to say something about properties of a matrix with positive nullity. As the comment on your post suggests, the Rank-Nullity theorem does the trick for this task.






share|cite|improve this answer





















  • Could you expand on these and give examples? Sorry, I am pretty confused about how to get the answers for these...
    – Thomas
    Nov 28 '18 at 5:33











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
});


}
});














draft saved

draft discarded


















StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3016660%2fconfused-with-linear-algebra-proofs%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









1














First, the dimensions of the matrices are irrelevant except for the fact that we want $AB$ to be a well defined matrix product. So you can safely ignore that for the moment.



You want to prove that $null(B)subset null(AB)$. So pick a vector in $null(B)$, say $vec{v}$. What can you say about the matrix-vector product $(AB)vec{v}$?



You also want to say something about properties of a matrix with positive nullity. As the comment on your post suggests, the Rank-Nullity theorem does the trick for this task.






share|cite|improve this answer





















  • Could you expand on these and give examples? Sorry, I am pretty confused about how to get the answers for these...
    – Thomas
    Nov 28 '18 at 5:33
















1














First, the dimensions of the matrices are irrelevant except for the fact that we want $AB$ to be a well defined matrix product. So you can safely ignore that for the moment.



You want to prove that $null(B)subset null(AB)$. So pick a vector in $null(B)$, say $vec{v}$. What can you say about the matrix-vector product $(AB)vec{v}$?



You also want to say something about properties of a matrix with positive nullity. As the comment on your post suggests, the Rank-Nullity theorem does the trick for this task.






share|cite|improve this answer





















  • Could you expand on these and give examples? Sorry, I am pretty confused about how to get the answers for these...
    – Thomas
    Nov 28 '18 at 5:33














1












1








1






First, the dimensions of the matrices are irrelevant except for the fact that we want $AB$ to be a well defined matrix product. So you can safely ignore that for the moment.



You want to prove that $null(B)subset null(AB)$. So pick a vector in $null(B)$, say $vec{v}$. What can you say about the matrix-vector product $(AB)vec{v}$?



You also want to say something about properties of a matrix with positive nullity. As the comment on your post suggests, the Rank-Nullity theorem does the trick for this task.






share|cite|improve this answer












First, the dimensions of the matrices are irrelevant except for the fact that we want $AB$ to be a well defined matrix product. So you can safely ignore that for the moment.



You want to prove that $null(B)subset null(AB)$. So pick a vector in $null(B)$, say $vec{v}$. What can you say about the matrix-vector product $(AB)vec{v}$?



You also want to say something about properties of a matrix with positive nullity. As the comment on your post suggests, the Rank-Nullity theorem does the trick for this task.







share|cite|improve this answer












share|cite|improve this answer



share|cite|improve this answer










answered Nov 28 '18 at 3:23









Valborg

542




542












  • Could you expand on these and give examples? Sorry, I am pretty confused about how to get the answers for these...
    – Thomas
    Nov 28 '18 at 5:33


















  • Could you expand on these and give examples? Sorry, I am pretty confused about how to get the answers for these...
    – Thomas
    Nov 28 '18 at 5:33
















Could you expand on these and give examples? Sorry, I am pretty confused about how to get the answers for these...
– Thomas
Nov 28 '18 at 5:33




Could you expand on these and give examples? Sorry, I am pretty confused about how to get the answers for these...
– Thomas
Nov 28 '18 at 5:33


















draft saved

draft discarded




















































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.




draft saved


draft discarded














StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3016660%2fconfused-with-linear-algebra-proofs%23new-answer', 'question_page');
}
);

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







-

Popular posts from this blog

Ellipse (mathématiques)

Image
Pour les articles homonymes, voir Ellipse. L'ellipse est le lieu des points dont la somme des distances à deux points fixes, dits foyers, est constante. Section du cône ou projection du cercle. En géométrie, une ellipse est une courbe plane fermée obtenue par l’intersection d’un cône de révolution avec un plan, à condition que celui-ci coupe l'axe de rotation du cône ou du cylindre : c'est une conique d'excentricité strictement comprise entre 0 et 1. On peut également la définir comme le lieu des points dont la somme des distances à deux points fixes, dits foyers, est constante (sa construction par la méthode du jardinier est très simple). Dans la vie courante, l’ellipse est la forme qu'on perçoit en regardant un cercle en perspective, ou la figure formée par l’ombre d'un disque sur une surface plane. On retrouve aussi, en première approximation [ 1 ] , des ellipses dans les trajectoires des corps célestes (planètes, comètes ou satellites arti...
Read more

Quarter-circle Tiles

Image
up vote 0 down vote favorite I have $99$ identical square tiles, each with a quarter-circle drawn on it like this: [asy] size(1.5cm); draw(Arc((2,0),1,90,180),red+1); draw((0,0)--(2,0)--(2,2)--(0,2)--(0,0)); [/asy] When I arrange the tiles in a $9times 11$ rectangular grid, each with a random orientation, what is the expected value of the number of full circles I form? I think this problem has to do with finding the chance any given 2x2 square has a circle, but I can't find it. expected-value share | cite | improve this question asked Nov 20 at 15:03 6minecraftninja 1 2 ...
Read more

Mont Emei

Image
Mont Emei.mw-parser-output .entete.map{background-image:url("//upload.wikimedia.org/wikipedia/commons/7/7a/Picto_infobox_map.png")} Vue du sommet Wanfo du mont Emei Géographie Altitude 3 099  m , Wanfo Coordonnées 29° 31′ 11″ nord, 103° 19′ 57″ est Administration Pays   Chine Province Sichuan Ville-préfecture Leshan Géolocalisation sur la carte : Sichuan Mont Emei Géolocalisation sur la carte : Chine Mont Emei modifier   Paysage panoramique du mont Emei, incluant le paysage panoramique du grand Bouddha de Leshan *   Patrimoine mondial de l'UNESCO Passerelle en bois sur le versant ouest Pays   Chine Subdivision Sichuan Type Mixte Critères (iv) (vi) (x) Superficie 15 400 ha Numéro d’identification 779 Zone géographique Asie et Pacifique  ** Année d’inscription 1996 (20 e session) * Descriptif officiel UNESCO ** Classification géog...
Read more