Why are transformations always on the left side of the object?
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I'm doing some philosophy involving time evolution operators. Thus I might have two operators $A$ and $B$ which operate in order on the state of the world, $x$. This might be written mathematically as $B(A(x))$ or in the multiplicative notation for groups and semigroups, $BAx$.
For time evolution, it is intuitive to write $xAB$, as time evolves from left to right. This is intuitive purely because of culture. Obviously both are equally valid. I merely need to define the operators with reversed multiplication tables.
Why do we choose to define our transformation operators to always appear on the left side of the multiplication? Is it merely because that's the order they appear in function composition, or is there something deeper?
functions convention
asked Dec 14 '18 at 17:59
Cort AmmonCort Ammon
2,381616
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add a comment |
$begingroup$
I'm doing some philosophy involving time evolution operators. Thus I might have two operators $A$ and $B$ which operate in order on the state of the world, $x$. This might be written mathematically as $B(A(x))$ or in the multiplicative notation for groups and semigroups, $BAx$.
For time evolution, it is intuitive to write $xAB$, as time evolves from left to right. This is intuitive purely because of culture. Obviously both are equally valid. I merely need to define the operators with reversed multiplication tables.
Why do we choose to define our transformation operators to always appear on the left side of the multiplication? Is it merely because that's the order they appear in function composition, or is there something deeper?
functions convention
asked Dec 14 '18 at 17:59
Cort AmmonCort Ammon
2,381616
$endgroup$
2
$begingroup$
Perhaps there is no deeper reason than "because Euler said so".
$endgroup$
– Mike Earnest
Dec 14 '18 at 18:24
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It’s not just in linear algebra that switching the order might be useful. The composition of two functions in general is quite confusing, since in $f circ g(x)$ you first apply $g$. In my opinion this makes things like reading off commutative diagrams much more of a hassle than it should be.
$endgroup$
– SvanN
Dec 14 '18 at 18:46
add a comment |
$begingroup$
I'm doing some philosophy involving time evolution operators. Thus I might have two operators $A$ and $B$ which operate in order on the state of the world, $x$. This might be written mathematically as $B(A(x))$ or in the multiplicative notation for groups and semigroups, $BAx$.
For time evolution, it is intuitive to write $xAB$, as time evolves from left to right. This is intuitive purely because of culture. Obviously both are equally valid. I merely need to define the operators with reversed multiplication tables.
Why do we choose to define our transformation operators to always appear on the left side of the multiplication? Is it merely because that's the order they appear in function composition, or is there something deeper?
functions convention
asked Dec 14 '18 at 17:59
Cort AmmonCort Ammon
2,381616
$endgroup$
I'm doing some philosophy involving time evolution operators. Thus I might have two operators $A$ and $B$ which operate in order on the state of the world, $x$. This might be written mathematically as $B(A(x))$ or in the multiplicative notation for groups and semigroups, $BAx$.
For time evolution, it is intuitive to write $xAB$, as time evolves from left to right. This is intuitive purely because of culture. Obviously both are equally valid. I merely need to define the operators with reversed multiplication tables.
Why do we choose to define our transformation operators to always appear on the left side of the multiplication? Is it merely because that's the order they appear in function composition, or is there something deeper?
functions convention
functions convention
asked Dec 14 '18 at 17:59
Cort AmmonCort Ammon
2,381616
asked Dec 14 '18 at 17:59
Cort AmmonCort Ammon
2,381616
asked Dec 14 '18 at 17:59
Cort AmmonCort Ammon
2,381616
asked Dec 14 '18 at 17:59
Cort AmmonCort Ammon
2,381616
asked Dec 14 '18 at 17:59
Cort AmmonCort Ammon
2,381616
2,381616
2
$begingroup$
Perhaps there is no deeper reason than "because Euler said so".
$endgroup$
– Mike Earnest
Dec 14 '18 at 18:24
$begingroup$
It’s not just in linear algebra that switching the order might be useful. The composition of two functions in general is quite confusing, since in $f circ g(x)$ you first apply $g$. In my opinion this makes things like reading off commutative diagrams much more of a hassle than it should be.
$endgroup$
– SvanN
Dec 14 '18 at 18:46
add a comment |
2
$begingroup$
Perhaps there is no deeper reason than "because Euler said so".
$endgroup$
– Mike Earnest
Dec 14 '18 at 18:24
$begingroup$
It’s not just in linear algebra that switching the order might be useful. The composition of two functions in general is quite confusing, since in $f circ g(x)$ you first apply $g$. In my opinion this makes things like reading off commutative diagrams much more of a hassle than it should be.
$endgroup$
– SvanN
Dec 14 '18 at 18:46
2
2
$begingroup$
Perhaps there is no deeper reason than "because Euler said so".
$endgroup$
– Mike Earnest
Dec 14 '18 at 18:24
$begingroup$
Perhaps there is no deeper reason than "because Euler said so".
$endgroup$
– Mike Earnest
Dec 14 '18 at 18:24
$begingroup$
It’s not just in linear algebra that switching the order might be useful. The composition of two functions in general is quite confusing, since in $f circ g(x)$ you first apply $g$. In my opinion this makes things like reading off commutative diagrams much more of a hassle than it should be.
$endgroup$
– SvanN
Dec 14 '18 at 18:46
$begingroup$
It’s not just in linear algebra that switching the order might be useful. The composition of two functions in general is quite confusing, since in $f circ g(x)$ you first apply $g$. In my opinion this makes things like reading off commutative diagrams much more of a hassle than it should be.
$endgroup$
– SvanN
Dec 14 '18 at 18:46
add a comment |
1 Answer
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The convention that operators go on the left is not universal. For example, Herstein's "Topics in Algebra" is written with operators on the right. See e.g.
this
answered Dec 14 '18 at 18:07
Robert IsraelRobert Israel
319k23209459
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add a comment |
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$begingroup$
The convention that operators go on the left is not universal. For example, Herstein's "Topics in Algebra" is written with operators on the right. See e.g.
this
answered Dec 14 '18 at 18:07
Robert IsraelRobert Israel
319k23209459
$endgroup$
add a comment |
$begingroup$
The convention that operators go on the left is not universal. For example, Herstein's "Topics in Algebra" is written with operators on the right. See e.g.
this
answered Dec 14 '18 at 18:07
Robert IsraelRobert Israel
319k23209459
$endgroup$
add a comment |
$begingroup$
The convention that operators go on the left is not universal. For example, Herstein's "Topics in Algebra" is written with operators on the right. See e.g.
this
answered Dec 14 '18 at 18:07
Robert IsraelRobert Israel
319k23209459
$endgroup$
The convention that operators go on the left is not universal. For example, Herstein's "Topics in Algebra" is written with operators on the right. See e.g.
this
answered Dec 14 '18 at 18:07
Robert IsraelRobert Israel
319k23209459
answered Dec 14 '18 at 18:07
Robert IsraelRobert Israel
319k23209459
answered Dec 14 '18 at 18:07
Robert IsraelRobert Israel
319k23209459
answered Dec 14 '18 at 18:07
Robert IsraelRobert Israel
319k23209459
319k23209459
add a comment |
add a comment |
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$begingroup$
Perhaps there is no deeper reason than "because Euler said so".
$endgroup$
– Mike Earnest
Dec 14 '18 at 18:24
$begingroup$
It’s not just in linear algebra that switching the order might be useful. The composition of two functions in general is quite confusing, since in $f circ g(x)$ you first apply $g$. In my opinion this makes things like reading off commutative diagrams much more of a hassle than it should be.
$endgroup$
– SvanN
Dec 14 '18 at 18:46