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




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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




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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




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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





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      asked Aug 20 at 10:23









      user6873235

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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.






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          • 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










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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.






          share|cite|improve this answer




















          • 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














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          1
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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.






          share|cite|improve this answer




















          • 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












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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.






          share|cite|improve this answer












          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.







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          answered Aug 20 at 11:50









          uniquesolution

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          • 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




          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












           

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