If differential operators are linear operators, what might it mean to act a differential operator to a function to its left?
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Given a differential operator like the regular derivative, or grad or curl or div etc, it can act on a function to its right to yield a new function. Because it is linear, it is effectively like a an infinite matrix acting on an infinite column vector (roughly, obviously the space of smooth functions in uncountable, +other differences).
But matrices can be applied to the other side on a row vector. What would be analogous for differential operators?
linear-algebra linear-transformations differential-operators
asked Aug 20 at 10:23
user6873235
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up vote
3
down vote
favorite
Given a differential operator like the regular derivative, or grad or curl or div etc, it can act on a function to its right to yield a new function. Because it is linear, it is effectively like a an infinite matrix acting on an infinite column vector (roughly, obviously the space of smooth functions in uncountable, +other differences).
But matrices can be applied to the other side on a row vector. What would be analogous for differential operators?
linear-algebra linear-transformations differential-operators
asked Aug 20 at 10:23
user6873235
47928
add a comment |Â
up vote
3
down vote
favorite
up vote
3
down vote
favorite
Given a differential operator like the regular derivative, or grad or curl or div etc, it can act on a function to its right to yield a new function. Because it is linear, it is effectively like a an infinite matrix acting on an infinite column vector (roughly, obviously the space of smooth functions in uncountable, +other differences).
But matrices can be applied to the other side on a row vector. What would be analogous for differential operators?
linear-algebra linear-transformations differential-operators
asked Aug 20 at 10:23
user6873235
47928
Given a differential operator like the regular derivative, or grad or curl or div etc, it can act on a function to its right to yield a new function. Because it is linear, it is effectively like a an infinite matrix acting on an infinite column vector (roughly, obviously the space of smooth functions in uncountable, +other differences).
But matrices can be applied to the other side on a row vector. What would be analogous for differential operators?
linear-algebra linear-transformations differential-operators
asked Aug 20 at 10:23
user6873235
47928
asked Aug 20 at 10:23
user6873235
47928
asked Aug 20 at 10:23
user6873235
47928
asked Aug 20 at 10:23
user6873235
47928
47928
add a comment |Â
add a comment |Â
1 Answer
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There is no natural "action on a function to its left" that can be assigned to a differential operator. For a linear operator, what matters is the result of its action; what does not matter is how we write it. The fact that we say "a matrix applied to other side on a row vector" is just our usage of the language. It does not create a new kind of action of a linear operator represented by the matrix.
answered Aug 20 at 11:50
uniquesolution
8,251823
But multiplying a matrix on a column vector to the right will give an entirely different result in general to application to an equivalent row vector.
– user6873235
Aug 21 at 12:43
add a comment |Â
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
1
down vote
There is no natural "action on a function to its left" that can be assigned to a differential operator. For a linear operator, what matters is the result of its action; what does not matter is how we write it. The fact that we say "a matrix applied to other side on a row vector" is just our usage of the language. It does not create a new kind of action of a linear operator represented by the matrix.
answered Aug 20 at 11:50
uniquesolution
8,251823
But multiplying a matrix on a column vector to the right will give an entirely different result in general to application to an equivalent row vector.
– user6873235
Aug 21 at 12:43
add a comment |Â
up vote
1
down vote
There is no natural "action on a function to its left" that can be assigned to a differential operator. For a linear operator, what matters is the result of its action; what does not matter is how we write it. The fact that we say "a matrix applied to other side on a row vector" is just our usage of the language. It does not create a new kind of action of a linear operator represented by the matrix.
answered Aug 20 at 11:50
uniquesolution
8,251823
But multiplying a matrix on a column vector to the right will give an entirely different result in general to application to an equivalent row vector.
– user6873235
Aug 21 at 12:43
add a comment |Â
up vote
1
down vote
up vote
1
down vote
There is no natural "action on a function to its left" that can be assigned to a differential operator. For a linear operator, what matters is the result of its action; what does not matter is how we write it. The fact that we say "a matrix applied to other side on a row vector" is just our usage of the language. It does not create a new kind of action of a linear operator represented by the matrix.
answered Aug 20 at 11:50
uniquesolution
8,251823
There is no natural "action on a function to its left" that can be assigned to a differential operator. For a linear operator, what matters is the result of its action; what does not matter is how we write it. The fact that we say "a matrix applied to other side on a row vector" is just our usage of the language. It does not create a new kind of action of a linear operator represented by the matrix.
answered Aug 20 at 11:50
uniquesolution
8,251823
answered Aug 20 at 11:50
uniquesolution
8,251823
answered Aug 20 at 11:50
uniquesolution
8,251823
answered Aug 20 at 11:50
uniquesolution
8,251823
8,251823
But multiplying a matrix on a column vector to the right will give an entirely different result in general to application to an equivalent row vector.
– user6873235
Aug 21 at 12:43
add a comment |Â
But multiplying a matrix on a column vector to the right will give an entirely different result in general to application to an equivalent row vector.
– user6873235
Aug 21 at 12:43
But multiplying a matrix on a column vector to the right will give an entirely different result in general to application to an equivalent row vector.
– user6873235
Aug 21 at 12:43
But multiplying a matrix on a column vector to the right will give an entirely different result in general to application to an equivalent row vector.
– user6873235
Aug 21 at 12:43
add a comment |Â
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