Mixing
In linear algebra, how do we define direct sum between two sets if we choose an expression like A + B = A +...
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in linear algebra, we define the direct sum between two sets like this :
A +* B = if and only if (A + B = C and A ∩ B = {0}).
How do we define A +* B = A +* B ? I mean : is it always true or only when A ∩ B = {0} ? In this case, that mean that A * B = A * B in general is not always true ?
Thanks`
linear-algebra
asked Jan 19 at 15:26
John HammondJohn Hammond
1
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closed as unclear what you're asking by Dietrich Burde, David C. Ullrich, Lord Shark the Unknown, Pacciu, Nosrati Jan 20 at 11:37
Please clarify your specific problem or add additional details to highlight exactly what you need. As it's currently written, it’s hard to tell exactly what you're asking. 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.
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$begingroup$
in linear algebra, we define the direct sum between two sets like this :
A +* B = if and only if (A + B = C and A ∩ B = {0}).
How do we define A +* B = A +* B ? I mean : is it always true or only when A ∩ B = {0} ? In this case, that mean that A * B = A * B in general is not always true ?
Thanks`
linear-algebra
asked Jan 19 at 15:26
John HammondJohn Hammond
1
$endgroup$
closed as unclear what you're asking by Dietrich Burde, David C. Ullrich, Lord Shark the Unknown, Pacciu, Nosrati Jan 20 at 11:37
Please clarify your specific problem or add additional details to highlight exactly what you need. As it's currently written, it’s hard to tell exactly what you're asking. 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.
1
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Is what always true? We can always define $A + B = lbrace a + b,vert, ain A,bin Brbrace$ (if this is what you mean) for subsets $A,Bsubset V$ (where $V$ is a vector space over some field $k$). And yes, this operation is well-defined: $A + B = A + B$. What distinguishes a direct sum (written $Aoplus B$, or perhaps A +* B in your notation) from a (regular) sum $A + B$ is that (i) we require that $A$ and $B$ are subspaces of $V$ and (ii) we require that $Acap B = lbrace 0rbrace$.
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– o.h.
Jan 19 at 15:30
$begingroup$
A +* B = A +* B
$endgroup$
– John Hammond
Jan 19 at 15:31
add a comment |
$begingroup$
in linear algebra, we define the direct sum between two sets like this :
A +* B = if and only if (A + B = C and A ∩ B = {0}).
How do we define A +* B = A +* B ? I mean : is it always true or only when A ∩ B = {0} ? In this case, that mean that A * B = A * B in general is not always true ?
Thanks`
linear-algebra
asked Jan 19 at 15:26
John HammondJohn Hammond
1
$endgroup$
in linear algebra, we define the direct sum between two sets like this :
A +* B = if and only if (A + B = C and A ∩ B = {0}).
How do we define A +* B = A +* B ? I mean : is it always true or only when A ∩ B = {0} ? In this case, that mean that A * B = A * B in general is not always true ?
Thanks`
linear-algebra
linear-algebra
asked Jan 19 at 15:26
John HammondJohn Hammond
1
asked Jan 19 at 15:26
John HammondJohn Hammond
1
asked Jan 19 at 15:26
John HammondJohn Hammond
1
asked Jan 19 at 15:26
John HammondJohn Hammond
1
asked Jan 19 at 15:26
John HammondJohn Hammond
1
1
closed as unclear what you're asking by Dietrich Burde, David C. Ullrich, Lord Shark the Unknown, Pacciu, Nosrati Jan 20 at 11:37
Please clarify your specific problem or add additional details to highlight exactly what you need. As it's currently written, it’s hard to tell exactly what you're asking. 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 unclear what you're asking by Dietrich Burde, David C. Ullrich, Lord Shark the Unknown, Pacciu, Nosrati Jan 20 at 11:37
Please clarify your specific problem or add additional details to highlight exactly what you need. As it's currently written, it’s hard to tell exactly what you're asking. 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.
1
$begingroup$
Is what always true? We can always define $A + B = lbrace a + b,vert, ain A,bin Brbrace$ (if this is what you mean) for subsets $A,Bsubset V$ (where $V$ is a vector space over some field $k$). And yes, this operation is well-defined: $A + B = A + B$. What distinguishes a direct sum (written $Aoplus B$, or perhaps A +* B in your notation) from a (regular) sum $A + B$ is that (i) we require that $A$ and $B$ are subspaces of $V$ and (ii) we require that $Acap B = lbrace 0rbrace$.
$endgroup$
– o.h.
Jan 19 at 15:30
$begingroup$
A +* B = A +* B
$endgroup$
– John Hammond
Jan 19 at 15:31
add a comment |
1
$begingroup$
Is what always true? We can always define $A + B = lbrace a + b,vert, ain A,bin Brbrace$ (if this is what you mean) for subsets $A,Bsubset V$ (where $V$ is a vector space over some field $k$). And yes, this operation is well-defined: $A + B = A + B$. What distinguishes a direct sum (written $Aoplus B$, or perhaps A +* B in your notation) from a (regular) sum $A + B$ is that (i) we require that $A$ and $B$ are subspaces of $V$ and (ii) we require that $Acap B = lbrace 0rbrace$.
$endgroup$
– o.h.
Jan 19 at 15:30
$begingroup$
A +* B = A +* B
$endgroup$
– John Hammond
Jan 19 at 15:31
1
1
$begingroup$
Is what always true? We can always define $A + B = lbrace a + b,vert, ain A,bin Brbrace$ (if this is what you mean) for subsets $A,Bsubset V$ (where $V$ is a vector space over some field $k$). And yes, this operation is well-defined: $A + B = A + B$. What distinguishes a direct sum (written $Aoplus B$, or perhaps A +* B in your notation) from a (regular) sum $A + B$ is that (i) we require that $A$ and $B$ are subspaces of $V$ and (ii) we require that $Acap B = lbrace 0rbrace$.
$endgroup$
– o.h.
Jan 19 at 15:30
$begingroup$
Is what always true? We can always define $A + B = lbrace a + b,vert, ain A,bin Brbrace$ (if this is what you mean) for subsets $A,Bsubset V$ (where $V$ is a vector space over some field $k$). And yes, this operation is well-defined: $A + B = A + B$. What distinguishes a direct sum (written $Aoplus B$, or perhaps A +* B in your notation) from a (regular) sum $A + B$ is that (i) we require that $A$ and $B$ are subspaces of $V$ and (ii) we require that $Acap B = lbrace 0rbrace$.
$endgroup$
– o.h.
Jan 19 at 15:30
$begingroup$
A +* B = A +* B
$endgroup$
– John Hammond
Jan 19 at 15:31
$begingroup$
A +* B = A +* B
$endgroup$
– John Hammond
Jan 19 at 15:31
add a comment |
1 Answer
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In general it is not very meaningful to ask whether some object is equal to itself. However, a question that often appears is whether a definition makes sense (i.e. if some object is well-defined), insofar as it does not depend on arbitrary choices. If we are trying to show that some object (e.g. function) $T$ is well-defined, this is sometimes written $T = T$. But really that's an abuse of notation.
If $A,Bsubset C$ and $A + B = C$ (every element of $C$ can be written as a sum of elements of $A$ and $B$), then we write $C = Aoplus B$ if and only if $Acap B = lbrace 0rbrace$. The $oplus$-notation is a kind of shorthand for a sum of subspaces having this intersection property. We don't want to write "let $A,Bsubset C$ st. $A + B = C$ and $Acap B = 0$" everytime this situation occurs, because (a) it occurs often and (b) it has very important consequences (e.g. every element of $C$ can be written uniquely as a sum of elements of $A$ and $B$).
answered Jan 19 at 15:39
o.h.o.h.
4616
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add a comment |
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
In general it is not very meaningful to ask whether some object is equal to itself. However, a question that often appears is whether a definition makes sense (i.e. if some object is well-defined), insofar as it does not depend on arbitrary choices. If we are trying to show that some object (e.g. function) $T$ is well-defined, this is sometimes written $T = T$. But really that's an abuse of notation.
If $A,Bsubset C$ and $A + B = C$ (every element of $C$ can be written as a sum of elements of $A$ and $B$), then we write $C = Aoplus B$ if and only if $Acap B = lbrace 0rbrace$. The $oplus$-notation is a kind of shorthand for a sum of subspaces having this intersection property. We don't want to write "let $A,Bsubset C$ st. $A + B = C$ and $Acap B = 0$" everytime this situation occurs, because (a) it occurs often and (b) it has very important consequences (e.g. every element of $C$ can be written uniquely as a sum of elements of $A$ and $B$).
answered Jan 19 at 15:39
o.h.o.h.
4616
$endgroup$
add a comment |
$begingroup$
In general it is not very meaningful to ask whether some object is equal to itself. However, a question that often appears is whether a definition makes sense (i.e. if some object is well-defined), insofar as it does not depend on arbitrary choices. If we are trying to show that some object (e.g. function) $T$ is well-defined, this is sometimes written $T = T$. But really that's an abuse of notation.
If $A,Bsubset C$ and $A + B = C$ (every element of $C$ can be written as a sum of elements of $A$ and $B$), then we write $C = Aoplus B$ if and only if $Acap B = lbrace 0rbrace$. The $oplus$-notation is a kind of shorthand for a sum of subspaces having this intersection property. We don't want to write "let $A,Bsubset C$ st. $A + B = C$ and $Acap B = 0$" everytime this situation occurs, because (a) it occurs often and (b) it has very important consequences (e.g. every element of $C$ can be written uniquely as a sum of elements of $A$ and $B$).
answered Jan 19 at 15:39
o.h.o.h.
4616
$endgroup$
add a comment |
$begingroup$
In general it is not very meaningful to ask whether some object is equal to itself. However, a question that often appears is whether a definition makes sense (i.e. if some object is well-defined), insofar as it does not depend on arbitrary choices. If we are trying to show that some object (e.g. function) $T$ is well-defined, this is sometimes written $T = T$. But really that's an abuse of notation.
If $A,Bsubset C$ and $A + B = C$ (every element of $C$ can be written as a sum of elements of $A$ and $B$), then we write $C = Aoplus B$ if and only if $Acap B = lbrace 0rbrace$. The $oplus$-notation is a kind of shorthand for a sum of subspaces having this intersection property. We don't want to write "let $A,Bsubset C$ st. $A + B = C$ and $Acap B = 0$" everytime this situation occurs, because (a) it occurs often and (b) it has very important consequences (e.g. every element of $C$ can be written uniquely as a sum of elements of $A$ and $B$).
answered Jan 19 at 15:39
o.h.o.h.
4616
$endgroup$
In general it is not very meaningful to ask whether some object is equal to itself. However, a question that often appears is whether a definition makes sense (i.e. if some object is well-defined), insofar as it does not depend on arbitrary choices. If we are trying to show that some object (e.g. function) $T$ is well-defined, this is sometimes written $T = T$. But really that's an abuse of notation.
If $A,Bsubset C$ and $A + B = C$ (every element of $C$ can be written as a sum of elements of $A$ and $B$), then we write $C = Aoplus B$ if and only if $Acap B = lbrace 0rbrace$. The $oplus$-notation is a kind of shorthand for a sum of subspaces having this intersection property. We don't want to write "let $A,Bsubset C$ st. $A + B = C$ and $Acap B = 0$" everytime this situation occurs, because (a) it occurs often and (b) it has very important consequences (e.g. every element of $C$ can be written uniquely as a sum of elements of $A$ and $B$).
answered Jan 19 at 15:39
o.h.o.h.
4616
edited Jan 19 at 15:49
answered Jan 19 at 15:39
o.h.o.h.
4616
answered Jan 19 at 15:39
o.h.o.h.
4616
answered Jan 19 at 15:39
o.h.o.h.
4616
4616
add a comment |
add a comment |
1
$begingroup$
Is what always true? We can always define $A + B = lbrace a + b,vert, ain A,bin Brbrace$ (if this is what you mean) for subsets $A,Bsubset V$ (where $V$ is a vector space over some field $k$). And yes, this operation is well-defined: $A + B = A + B$. What distinguishes a direct sum (written $Aoplus B$, or perhaps A +* B in your notation) from a (regular) sum $A + B$ is that (i) we require that $A$ and $B$ are subspaces of $V$ and (ii) we require that $Acap B = lbrace 0rbrace$.
$endgroup$
– o.h.
Jan 19 at 15:30
$begingroup$
A +* B = A +* B
$endgroup$
– John Hammond
Jan 19 at 15:31