Silly question about the definition of Sum of Subspaces

Multi tool use
Multi tool use

The name of the pictureThe name of the pictureThe name of the pictureClash Royale CLAN TAG#URR8PPP











up vote
0
down vote

favorite













Definition: Let $mathcalV$ be a vector space over the field $mathbbR$ and $mathcalU$ and $mathcalW$ subspaces of $mathcalV$.



The set:



$$mathcalU+mathcalW := u boxplus w : uin mathcalU,win mathcalW $$



Is called the SUM of subspaces $mathcalU$ and $mathcalW$.




My question is, obviously the element $u boxplus w$ have a notion of sum given by the definition of $boxplus$. But before we prove if the SUM is indeed a subspace of $mathcalV$, the binary operation $boxplus$ is the operation defined in $mathcalV$? I mean $u boxplus w equiv u boxplus_mathcalV w$ ?



(*) The symbol $boxplus$ is the vector adition. And the other symbol $boxplus_mathcalV$ is the vector adition defined specifically in the vector space $mathcalV$






linear-algebra abstract-algebra







share|cite|improve this question



















  • 4




    What is $boxplus$? If not provided, then we cannot know.
    – xbh
    Sep 2 at 9:19











  • Is a notion of vector adition. But I don't know if is the same notion of vector adition defined in $mathcalV$.
    – Jack Clerk
    Sep 2 at 9:21






  • 1




    You are right. The addition is just the addition on $V$.
    – James
    Sep 2 at 9:22






  • 1




    @JackClerk Actually I don't know, cause symbols without clarification could mean anything. But I may guess this is just the addition on $V$ [but I cannot say for sure, maybe check your textbook].
    – xbh
    Sep 2 at 9:23






  • 1




    Just the addition of vectors in $V$. Well-defined since $U$ and $W$ are subspaces of $V$.
    – Wuestenfux
    Sep 2 at 9:27














up vote
0
down vote

favorite













Definition: Let $mathcalV$ be a vector space over the field $mathbbR$ and $mathcalU$ and $mathcalW$ subspaces of $mathcalV$.



The set:



$$mathcalU+mathcalW := u boxplus w : uin mathcalU,win mathcalW $$



Is called the SUM of subspaces $mathcalU$ and $mathcalW$.




My question is, obviously the element $u boxplus w$ have a notion of sum given by the definition of $boxplus$. But before we prove if the SUM is indeed a subspace of $mathcalV$, the binary operation $boxplus$ is the operation defined in $mathcalV$? I mean $u boxplus w equiv u boxplus_mathcalV w$ ?



(*) The symbol $boxplus$ is the vector adition. And the other symbol $boxplus_mathcalV$ is the vector adition defined specifically in the vector space $mathcalV$






linear-algebra abstract-algebra







share|cite|improve this question



















  • 4




    What is $boxplus$? If not provided, then we cannot know.
    – xbh
    Sep 2 at 9:19











  • Is a notion of vector adition. But I don't know if is the same notion of vector adition defined in $mathcalV$.
    – Jack Clerk
    Sep 2 at 9:21






  • 1




    You are right. The addition is just the addition on $V$.
    – James
    Sep 2 at 9:22






  • 1




    @JackClerk Actually I don't know, cause symbols without clarification could mean anything. But I may guess this is just the addition on $V$ [but I cannot say for sure, maybe check your textbook].
    – xbh
    Sep 2 at 9:23






  • 1




    Just the addition of vectors in $V$. Well-defined since $U$ and $W$ are subspaces of $V$.
    – Wuestenfux
    Sep 2 at 9:27












up vote
0
down vote

favorite









up vote
0
down vote

favorite












Definition: Let $mathcalV$ be a vector space over the field $mathbbR$ and $mathcalU$ and $mathcalW$ subspaces of $mathcalV$.



The set:



$$mathcalU+mathcalW := u boxplus w : uin mathcalU,win mathcalW $$



Is called the SUM of subspaces $mathcalU$ and $mathcalW$.




My question is, obviously the element $u boxplus w$ have a notion of sum given by the definition of $boxplus$. But before we prove if the SUM is indeed a subspace of $mathcalV$, the binary operation $boxplus$ is the operation defined in $mathcalV$? I mean $u boxplus w equiv u boxplus_mathcalV w$ ?



(*) The symbol $boxplus$ is the vector adition. And the other symbol $boxplus_mathcalV$ is the vector adition defined specifically in the vector space $mathcalV$






linear-algebra abstract-algebra







share|cite|improve this question
















Definition: Let $mathcalV$ be a vector space over the field $mathbbR$ and $mathcalU$ and $mathcalW$ subspaces of $mathcalV$.



The set:



$$mathcalU+mathcalW := u boxplus w : uin mathcalU,win mathcalW $$



Is called the SUM of subspaces $mathcalU$ and $mathcalW$.




My question is, obviously the element $u boxplus w$ have a notion of sum given by the definition of $boxplus$. But before we prove if the SUM is indeed a subspace of $mathcalV$, the binary operation $boxplus$ is the operation defined in $mathcalV$? I mean $u boxplus w equiv u boxplus_mathcalV w$ ?



(*) The symbol $boxplus$ is the vector adition. And the other symbol $boxplus_mathcalV$ is the vector adition defined specifically in the vector space $mathcalV$






linear-algebra abstract-algebra




linear-algebra abstract-algebra






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited Sep 2 at 9:27

























asked Sep 2 at 9:17









Jack Clerk

707




707







  • 4




    What is $boxplus$? If not provided, then we cannot know.
    – xbh
    Sep 2 at 9:19











  • Is a notion of vector adition. But I don't know if is the same notion of vector adition defined in $mathcalV$.
    – Jack Clerk
    Sep 2 at 9:21






  • 1




    You are right. The addition is just the addition on $V$.
    – James
    Sep 2 at 9:22






  • 1




    @JackClerk Actually I don't know, cause symbols without clarification could mean anything. But I may guess this is just the addition on $V$ [but I cannot say for sure, maybe check your textbook].
    – xbh
    Sep 2 at 9:23






  • 1




    Just the addition of vectors in $V$. Well-defined since $U$ and $W$ are subspaces of $V$.
    – Wuestenfux
    Sep 2 at 9:27












  • 4




    What is $boxplus$? If not provided, then we cannot know.
    – xbh
    Sep 2 at 9:19











  • Is a notion of vector adition. But I don't know if is the same notion of vector adition defined in $mathcalV$.
    – Jack Clerk
    Sep 2 at 9:21






  • 1




    You are right. The addition is just the addition on $V$.
    – James
    Sep 2 at 9:22






  • 1




    @JackClerk Actually I don't know, cause symbols without clarification could mean anything. But I may guess this is just the addition on $V$ [but I cannot say for sure, maybe check your textbook].
    – xbh
    Sep 2 at 9:23






  • 1




    Just the addition of vectors in $V$. Well-defined since $U$ and $W$ are subspaces of $V$.
    – Wuestenfux
    Sep 2 at 9:27







4




4




What is $boxplus$? If not provided, then we cannot know.
– xbh
Sep 2 at 9:19





What is $boxplus$? If not provided, then we cannot know.
– xbh
Sep 2 at 9:19













Is a notion of vector adition. But I don't know if is the same notion of vector adition defined in $mathcalV$.
– Jack Clerk
Sep 2 at 9:21




Is a notion of vector adition. But I don't know if is the same notion of vector adition defined in $mathcalV$.
– Jack Clerk
Sep 2 at 9:21




1




1




You are right. The addition is just the addition on $V$.
– James
Sep 2 at 9:22




You are right. The addition is just the addition on $V$.
– James
Sep 2 at 9:22




1




1




@JackClerk Actually I don't know, cause symbols without clarification could mean anything. But I may guess this is just the addition on $V$ [but I cannot say for sure, maybe check your textbook].
– xbh
Sep 2 at 9:23




@JackClerk Actually I don't know, cause symbols without clarification could mean anything. But I may guess this is just the addition on $V$ [but I cannot say for sure, maybe check your textbook].
– xbh
Sep 2 at 9:23




1




1




Just the addition of vectors in $V$. Well-defined since $U$ and $W$ are subspaces of $V$.
– Wuestenfux
Sep 2 at 9:27




Just the addition of vectors in $V$. Well-defined since $U$ and $W$ are subspaces of $V$.
– Wuestenfux
Sep 2 at 9:27















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',
convertImagesToLinks: true,
noModals: false,
showLowRepImageUploadWarning: true,
reputationToPostImages: 10,
bindNavPrevention: true,
postfix: "",
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%2f2902526%2fsilly-question-about-the-definition-of-sum-of-subspaces%23new-answer', 'question_page');

);

Post as a guest






















active

oldest

votes













active

oldest

votes









active

oldest

votes






active

oldest

votes















 

draft saved


draft discarded















































 


draft saved


draft discarded














StackExchange.ready(
function ()
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f2902526%2fsilly-question-about-the-definition-of-sum-of-subspaces%23new-answer', 'question_page');

);

Post as a guest
































































sP2LjJp63NIT3hAoFhCgWCL8mCdobKrrS8TZr7AB
-
Rlq1a,27kmwYDR3CF6LMY0eCuZ,GMcuBiPtxW12Xms7ZqQEe9R

這個網誌中的熱門文章

How to combine Bézier curves to a surface?

圖片
Clash Royale CLAN TAG #URR8PPP up vote 2 down vote favorite My aim is to smooth the terrain in a video game. Therefore I contrived an algorithm that makes use of Bézier curves of different orders. But this algorithm is defined in a two dimensional space for now. To shift it into the third dimension I need to somehow combine the Bézier curves. Given two curves in each two dimensions, how can I combine them to build a surface? I thought about something like a curve of a curve. Or maybe I could multiply them somehow. How can I combine them to cause the wanted behavior? Is there a common approach? In the image you can see the input, on the left, and the output, on the right, of the algorithm that works in a two dimensional space. For the real application there is a three dimensional input. The algorithm relays only on the adjacent tiles. Given these it will define the control points of the mentioned Bézier cur...
閱讀完整內容

Propositional logic and tautologies

圖片
Clash Royale CLAN TAG #URR8PPP up vote 0 down vote favorite Taken from P. Suppes "Introduction to logic" pp16 Starting from the sentence "If $P$ tautologically implies $Q$, then.." Question: Is it true that, at least within propositional logic, we are able to prove only tautologies or do I mis-understand something? If yes, is propositional logic special in some way? Is the notion of tautology meaningful only in propositional logic? logic propositional-calculus share | cite | improve this question edited Nov 29 '17 at 18:53 Mauro ALLEGRANZA 60.8k 4 46 105 asked Nov 29 '17 at 16:32 Alvin Lepik 2,448 9 20 add a comment  |  up vote 0 down vote favorite Taken from P. Suppes "Introduction to logic" pp16 Starting from the sentence "If $P$ tautologically implies $Q$, then.." Question: Is it true that, at least within propositional logic, we are able to prove only tautologi...
閱讀完整內容

Distribution of Stopped Wiener Process with Stochastic Volatility

圖片
Clash Royale CLAN TAG #URR8PPP up vote 0 down vote favorite Let $(W_s)_s geq 0$ be a Wiener process and $tau$ be a random variable with an exponential distribution with parameter $lambda$. Suppose that $W$ and $tau$ are independent. In this question, we see that the distribution of the stopped Wiener process $W_tau$ corresponds to a Laplace distribution with scale parameter $fracsigmasqrt2lambda$ where $sigma$ is the instantaneous variance of the Wiener process. Suppose now, that the variance $sigma^2$ also follows a stochastic process such that we have: $$dS = sigma SdW_s \ dsigma^2 = alphasigma^2dt + xisigma^2dW_sigma$$ where $alpha$ and $xi$ are independent of $S$ and $dW_s$ and $dW_sigma$ are independent Wiener processes. My aim is to derive the distribution of $log S_tau$. First, if $barV_T$ denotes the mean variance over some time interval $[0,T]$ defined by $$barV_T = frac1T intlimits_0^Tsigma^2(t)dt,$$ it is easy to show (Lemma of Îto) that $$...
閱讀完整內容