Defining addition for vector spaces [closed] The 2019 Stack Overflow Developer Survey Results Are In Announcing the arrival of Valued Associate #679: Cesar Manara Planned maintenance scheduled April 17/18, 2019 at 00:00UTC (8:00pm US/Eastern)List of various vector (linear) spacesIsomorphism between 2 vector spacesExamples of 'almost' vector spaces where unitary law failsVector Spaces and GroupsAddition of Vector SpacesPrecisely defining complex vector spacesAddition between subspaces from different vectorspacesdo all vector spaces function under ordinary component wise addition?Properties of vector spaces.Subspaces and elements of isomorphic vector spaces
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Defining addition for vector spaces [closed]
The 2019 Stack Overflow Developer Survey Results Are In
Announcing the arrival of Valued Associate #679: Cesar Manara
Planned maintenance scheduled April 17/18, 2019 at 00:00UTC (8:00pm US/Eastern)List of various vector (linear) spacesIsomorphism between 2 vector spacesExamples of 'almost' vector spaces where unitary law failsVector Spaces and GroupsAddition of Vector SpacesPrecisely defining complex vector spacesAddition between subspaces from different vectorspacesdo all vector spaces function under ordinary component wise addition?Properties of vector spaces.Subspaces and elements of isomorphic vector spaces
$begingroup$
I am constructing a vector space where the elements are matrices, and I am looking for specified subsets to be subspaces. With my addition operation being element-wise addition of the matrices, this does not hold.
So I plan to use a different addition operation. What are the limitations on how I can change the addition operation? Can I make it any pairwise operation?
linear-algebra vector-spaces
asked Apr 8 at 12:42
MeowBlingBlingMeowBlingBling
1226
$endgroup$
closed as too broad by 5xum, Adrian Keister, José Carlos Santos, A. Pongrácz, Mike Earnest Apr 8 at 21:52
Please edit the question to limit it to a specific problem with enough detail to identify an adequate answer. Avoid asking multiple distinct questions at once. See the How to Ask page for help clarifying this question. If this question can be reworded to fit the rules in the help center, please edit the question.
add a comment |
$begingroup$
I am constructing a vector space where the elements are matrices, and I am looking for specified subsets to be subspaces. With my addition operation being element-wise addition of the matrices, this does not hold.
So I plan to use a different addition operation. What are the limitations on how I can change the addition operation? Can I make it any pairwise operation?
linear-algebra vector-spaces
asked Apr 8 at 12:42
MeowBlingBlingMeowBlingBling
1226
$endgroup$
closed as too broad by 5xum, Adrian Keister, José Carlos Santos, A. Pongrácz, Mike Earnest Apr 8 at 21:52
Please edit the question to limit it to a specific problem with enough detail to identify an adequate answer. Avoid asking multiple distinct questions at once. See the How to Ask page for help clarifying this question. If this question can be reworded to fit the rules in the help center, please edit the question.
2
$begingroup$
The addition operation is essentially just a function from $Vtimes V$ to $V$, where $V$ is your vector space. Of course, it must also make sure to satisfy the axioms of vector spaces.
$endgroup$
– Minus One-Twelfth
Apr 8 at 12:44
$begingroup$
This is a little to broad for us to answer. Which subsets do you want to be subspaces?
$endgroup$
– 5xum
Apr 8 at 12:46
1
$begingroup$
@MinusOne-Twelfth that answers my questions, I need to look back to the axioms( commutativity, associativity, ...etc)
$endgroup$
– MeowBlingBling
Apr 8 at 12:56
add a comment |
$begingroup$
I am constructing a vector space where the elements are matrices, and I am looking for specified subsets to be subspaces. With my addition operation being element-wise addition of the matrices, this does not hold.
So I plan to use a different addition operation. What are the limitations on how I can change the addition operation? Can I make it any pairwise operation?
linear-algebra vector-spaces
asked Apr 8 at 12:42
MeowBlingBlingMeowBlingBling
1226
$endgroup$
I am constructing a vector space where the elements are matrices, and I am looking for specified subsets to be subspaces. With my addition operation being element-wise addition of the matrices, this does not hold.
So I plan to use a different addition operation. What are the limitations on how I can change the addition operation? Can I make it any pairwise operation?
linear-algebra vector-spaces
linear-algebra vector-spaces
asked Apr 8 at 12:42
MeowBlingBlingMeowBlingBling
1226
asked Apr 8 at 12:42
MeowBlingBlingMeowBlingBling
1226
asked Apr 8 at 12:42
MeowBlingBlingMeowBlingBling
1226
asked Apr 8 at 12:42
MeowBlingBlingMeowBlingBling
1226
asked Apr 8 at 12:42
MeowBlingBlingMeowBlingBling
1226
1226
closed as too broad by 5xum, Adrian Keister, José Carlos Santos, A. Pongrácz, Mike Earnest Apr 8 at 21:52
Please edit the question to limit it to a specific problem with enough detail to identify an adequate answer. Avoid asking multiple distinct questions at once. See the How to Ask page for help clarifying this question. If this question can be reworded to fit the rules in the help center, please edit the question.
closed as too broad by 5xum, Adrian Keister, José Carlos Santos, A. Pongrácz, Mike Earnest Apr 8 at 21:52
Please edit the question to limit it to a specific problem with enough detail to identify an adequate answer. Avoid asking multiple distinct questions at once. See the How to Ask page for help clarifying this question. If this question can be reworded to fit the rules in the help center, please edit the question.
2
$begingroup$
The addition operation is essentially just a function from $Vtimes V$ to $V$, where $V$ is your vector space. Of course, it must also make sure to satisfy the axioms of vector spaces.
$endgroup$
– Minus One-Twelfth
Apr 8 at 12:44
$begingroup$
This is a little to broad for us to answer. Which subsets do you want to be subspaces?
$endgroup$
– 5xum
Apr 8 at 12:46
1
$begingroup$
@MinusOne-Twelfth that answers my questions, I need to look back to the axioms( commutativity, associativity, ...etc)
$endgroup$
– MeowBlingBling
Apr 8 at 12:56
add a comment |
2
$begingroup$
The addition operation is essentially just a function from $Vtimes V$ to $V$, where $V$ is your vector space. Of course, it must also make sure to satisfy the axioms of vector spaces.
$endgroup$
– Minus One-Twelfth
Apr 8 at 12:44
$begingroup$
This is a little to broad for us to answer. Which subsets do you want to be subspaces?
$endgroup$
– 5xum
Apr 8 at 12:46
1
$begingroup$
@MinusOne-Twelfth that answers my questions, I need to look back to the axioms( commutativity, associativity, ...etc)
$endgroup$
– MeowBlingBling
Apr 8 at 12:56
2
2
$begingroup$
The addition operation is essentially just a function from $Vtimes V$ to $V$, where $V$ is your vector space. Of course, it must also make sure to satisfy the axioms of vector spaces.
$endgroup$
– Minus One-Twelfth
Apr 8 at 12:44
$begingroup$
The addition operation is essentially just a function from $Vtimes V$ to $V$, where $V$ is your vector space. Of course, it must also make sure to satisfy the axioms of vector spaces.
$endgroup$
– Minus One-Twelfth
Apr 8 at 12:44
$begingroup$
This is a little to broad for us to answer. Which subsets do you want to be subspaces?
$endgroup$
– 5xum
Apr 8 at 12:46
$begingroup$
This is a little to broad for us to answer. Which subsets do you want to be subspaces?
$endgroup$
– 5xum
Apr 8 at 12:46
1
1
$begingroup$
@MinusOne-Twelfth that answers my questions, I need to look back to the axioms( commutativity, associativity, ...etc)
$endgroup$
– MeowBlingBling
Apr 8 at 12:56
$begingroup$
@MinusOne-Twelfth that answers my questions, I need to look back to the axioms( commutativity, associativity, ...etc)
$endgroup$
– MeowBlingBling
Apr 8 at 12:56
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
You can define your addition operation how you like so long as it satisfies the vector space axioms for addition (commutativity, associativity, additive identity, additive inverse). See wolfram
So for any $x,y,zin V$ and operation $+$ the following should hold
$$x +y = y+x$$
$$(x+y)+z = x+(y+z)$$
There should exist an identity element $0$ such that
$$0+x = x+0 = x$$
There should exist an inverse element $-x$ such that
$$x + (-x) = 0$$
Note that the scalar multiplication must also satisfy some axioms, including distributivity of vector sums, distributivity of scalar sums, associativity, scalar multiplication identity.
answered Apr 8 at 14:41
MeowBlingBlingMeowBlingBling
1226
$endgroup$
add a comment |
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
You can define your addition operation how you like so long as it satisfies the vector space axioms for addition (commutativity, associativity, additive identity, additive inverse). See wolfram
So for any $x,y,zin V$ and operation $+$ the following should hold
$$x +y = y+x$$
$$(x+y)+z = x+(y+z)$$
There should exist an identity element $0$ such that
$$0+x = x+0 = x$$
There should exist an inverse element $-x$ such that
$$x + (-x) = 0$$
Note that the scalar multiplication must also satisfy some axioms, including distributivity of vector sums, distributivity of scalar sums, associativity, scalar multiplication identity.
answered Apr 8 at 14:41
MeowBlingBlingMeowBlingBling
1226
$endgroup$
add a comment |
$begingroup$
You can define your addition operation how you like so long as it satisfies the vector space axioms for addition (commutativity, associativity, additive identity, additive inverse). See wolfram
So for any $x,y,zin V$ and operation $+$ the following should hold
$$x +y = y+x$$
$$(x+y)+z = x+(y+z)$$
There should exist an identity element $0$ such that
$$0+x = x+0 = x$$
There should exist an inverse element $-x$ such that
$$x + (-x) = 0$$
Note that the scalar multiplication must also satisfy some axioms, including distributivity of vector sums, distributivity of scalar sums, associativity, scalar multiplication identity.
answered Apr 8 at 14:41
MeowBlingBlingMeowBlingBling
1226
$endgroup$
add a comment |
$begingroup$
You can define your addition operation how you like so long as it satisfies the vector space axioms for addition (commutativity, associativity, additive identity, additive inverse). See wolfram
So for any $x,y,zin V$ and operation $+$ the following should hold
$$x +y = y+x$$
$$(x+y)+z = x+(y+z)$$
There should exist an identity element $0$ such that
$$0+x = x+0 = x$$
There should exist an inverse element $-x$ such that
$$x + (-x) = 0$$
Note that the scalar multiplication must also satisfy some axioms, including distributivity of vector sums, distributivity of scalar sums, associativity, scalar multiplication identity.
answered Apr 8 at 14:41
MeowBlingBlingMeowBlingBling
1226
$endgroup$
You can define your addition operation how you like so long as it satisfies the vector space axioms for addition (commutativity, associativity, additive identity, additive inverse). See wolfram
So for any $x,y,zin V$ and operation $+$ the following should hold
$$x +y = y+x$$
$$(x+y)+z = x+(y+z)$$
There should exist an identity element $0$ such that
$$0+x = x+0 = x$$
There should exist an inverse element $-x$ such that
$$x + (-x) = 0$$
Note that the scalar multiplication must also satisfy some axioms, including distributivity of vector sums, distributivity of scalar sums, associativity, scalar multiplication identity.
answered Apr 8 at 14:41
MeowBlingBlingMeowBlingBling
1226
answered Apr 8 at 14:41
MeowBlingBlingMeowBlingBling
1226
answered Apr 8 at 14:41
MeowBlingBlingMeowBlingBling
1226
answered Apr 8 at 14:41
MeowBlingBlingMeowBlingBling
1226
1226
add a comment |
add a comment |
2
$begingroup$
The addition operation is essentially just a function from $Vtimes V$ to $V$, where $V$ is your vector space. Of course, it must also make sure to satisfy the axioms of vector spaces.
$endgroup$
– Minus One-Twelfth
Apr 8 at 12:44
$begingroup$
This is a little to broad for us to answer. Which subsets do you want to be subspaces?
$endgroup$
– 5xum
Apr 8 at 12:46
1
$begingroup$
@MinusOne-Twelfth that answers my questions, I need to look back to the axioms( commutativity, associativity, ...etc)
$endgroup$
– MeowBlingBling
Apr 8 at 12:56