Silly question about the definition of Sum of Subspaces

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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







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  • 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















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