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
asked Sep 2 at 9:17
Jack Clerk
707
 |Â
show 5 more comments
up vote
0
down vote
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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
asked Sep 2 at 9:17
Jack Clerk
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
 |Â
show 5 more comments
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
asked Sep 2 at 9:17
Jack Clerk
707
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
asked Sep 2 at 9:17
Jack Clerk
707
asked Sep 2 at 9:17
Jack Clerk
707
edited Sep 2 at 9:27
asked Sep 2 at 9:17
Jack Clerk
707
asked Sep 2 at 9:17
Jack Clerk
707
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
 |Â
show 5 more comments
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
 |Â
show 5 more comments
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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