Difference between kernel of a matrix and kernel of a transformation?












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Does the word "kernel" carry different meanings in the contexts of matrices and transformations? Or are the definitions of the two in those two contexts intertwined in some way?



I know the kernel of a matrix to be it's null space. I haven't been taught about the kernel of a transformation yet, but I looked it up and it's a bunch of complicated stuff that doesn't make much sense (it's apparently every vector v that results in a transformation T outputting the zero vector?)



Are they just two different ideas?






linear-algebra matrices linear-transformations definition







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  • $begingroup$
    See Kernel of a Linear transformation : The kernel of a linear transformation L is the set of all vectors v such that $ L(v) = 0$.
    $endgroup$
    – Mauro ALLEGRANZA
    Dec 15 '18 at 15:10


















0












$begingroup$


Does the word "kernel" carry different meanings in the contexts of matrices and transformations? Or are the definitions of the two in those two contexts intertwined in some way?



I know the kernel of a matrix to be it's null space. I haven't been taught about the kernel of a transformation yet, but I looked it up and it's a bunch of complicated stuff that doesn't make much sense (it's apparently every vector v that results in a transformation T outputting the zero vector?)



Are they just two different ideas?






linear-algebra matrices linear-transformations definition







share|cite|improve this question











$endgroup$












  • $begingroup$
    See Kernel of a Linear transformation : The kernel of a linear transformation L is the set of all vectors v such that $ L(v) = 0$.
    $endgroup$
    – Mauro ALLEGRANZA
    Dec 15 '18 at 15:10
















0












0








0





$begingroup$


Does the word "kernel" carry different meanings in the contexts of matrices and transformations? Or are the definitions of the two in those two contexts intertwined in some way?



I know the kernel of a matrix to be it's null space. I haven't been taught about the kernel of a transformation yet, but I looked it up and it's a bunch of complicated stuff that doesn't make much sense (it's apparently every vector v that results in a transformation T outputting the zero vector?)



Are they just two different ideas?






linear-algebra matrices linear-transformations definition







share|cite|improve this question











$endgroup$




Does the word "kernel" carry different meanings in the contexts of matrices and transformations? Or are the definitions of the two in those two contexts intertwined in some way?



I know the kernel of a matrix to be it's null space. I haven't been taught about the kernel of a transformation yet, but I looked it up and it's a bunch of complicated stuff that doesn't make much sense (it's apparently every vector v that results in a transformation T outputting the zero vector?)



Are they just two different ideas?






linear-algebra matrices linear-transformations definition




linear-algebra matrices linear-transformations definition






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited Dec 15 '18 at 15:27









José Carlos Santos

160k22127232




160k22127232










asked Dec 15 '18 at 15:04









James RonaldJames Ronald

1257




1257












  • $begingroup$
    See Kernel of a Linear transformation : The kernel of a linear transformation L is the set of all vectors v such that $ L(v) = 0$.
    $endgroup$
    – Mauro ALLEGRANZA
    Dec 15 '18 at 15:10




















  • $begingroup$
    See Kernel of a Linear transformation : The kernel of a linear transformation L is the set of all vectors v such that $ L(v) = 0$.
    $endgroup$
    – Mauro ALLEGRANZA
    Dec 15 '18 at 15:10


















$begingroup$
See Kernel of a Linear transformation : The kernel of a linear transformation L is the set of all vectors v such that $ L(v) = 0$.
$endgroup$
– Mauro ALLEGRANZA
Dec 15 '18 at 15:10






$begingroup$
See Kernel of a Linear transformation : The kernel of a linear transformation L is the set of all vectors v such that $ L(v) = 0$.
$endgroup$
– Mauro ALLEGRANZA
Dec 15 '18 at 15:10












2 Answers
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The kernel of a matrix $A$ is the space of those vectors $v$ such that $A.v=0$. The kernel of a linear transformation $T$ is the space of those vectors $v$ such that $T(v)=0$. So, it is basically the same thing.






share|cite|improve this answer











$endgroup$





















    1












    $begingroup$

    Every linear transformation from one finite dimensional space to another can be written as a matrix. The only difference between "kernel of a linear transformation" and "kernel of a matrix" would be in the case of a linear transformation over infinite dimensional spaces which cannot be written as a matrix. An example would be the "differentiation" operator on the space of all differentiable functions. The kernel of that operator is the subspace of all constant functions.






    share|cite|improve this answer









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      2 Answers
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      2 Answers
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      $begingroup$

      The kernel of a matrix $A$ is the space of those vectors $v$ such that $A.v=0$. The kernel of a linear transformation $T$ is the space of those vectors $v$ such that $T(v)=0$. So, it is basically the same thing.






      share|cite|improve this answer











      $endgroup$


















        2












        $begingroup$

        The kernel of a matrix $A$ is the space of those vectors $v$ such that $A.v=0$. The kernel of a linear transformation $T$ is the space of those vectors $v$ such that $T(v)=0$. So, it is basically the same thing.






        share|cite|improve this answer











        $endgroup$
















          2












          2








          2





          $begingroup$

          The kernel of a matrix $A$ is the space of those vectors $v$ such that $A.v=0$. The kernel of a linear transformation $T$ is the space of those vectors $v$ such that $T(v)=0$. So, it is basically the same thing.






          share|cite|improve this answer











          $endgroup$



          The kernel of a matrix $A$ is the space of those vectors $v$ such that $A.v=0$. The kernel of a linear transformation $T$ is the space of those vectors $v$ such that $T(v)=0$. So, it is basically the same thing.







          share|cite|improve this answer














          share|cite|improve this answer



          share|cite|improve this answer








          edited Dec 15 '18 at 15:18









          Ivo Terek

          46.2k953142




          46.2k953142










          answered Dec 15 '18 at 15:10









          José Carlos SantosJosé Carlos Santos

          160k22127232




          160k22127232























              1












              $begingroup$

              Every linear transformation from one finite dimensional space to another can be written as a matrix. The only difference between "kernel of a linear transformation" and "kernel of a matrix" would be in the case of a linear transformation over infinite dimensional spaces which cannot be written as a matrix. An example would be the "differentiation" operator on the space of all differentiable functions. The kernel of that operator is the subspace of all constant functions.






              share|cite|improve this answer









              $endgroup$


















                1












                $begingroup$

                Every linear transformation from one finite dimensional space to another can be written as a matrix. The only difference between "kernel of a linear transformation" and "kernel of a matrix" would be in the case of a linear transformation over infinite dimensional spaces which cannot be written as a matrix. An example would be the "differentiation" operator on the space of all differentiable functions. The kernel of that operator is the subspace of all constant functions.






                share|cite|improve this answer









                $endgroup$
















                  1












                  1








                  1





                  $begingroup$

                  Every linear transformation from one finite dimensional space to another can be written as a matrix. The only difference between "kernel of a linear transformation" and "kernel of a matrix" would be in the case of a linear transformation over infinite dimensional spaces which cannot be written as a matrix. An example would be the "differentiation" operator on the space of all differentiable functions. The kernel of that operator is the subspace of all constant functions.






                  share|cite|improve this answer









                  $endgroup$



                  Every linear transformation from one finite dimensional space to another can be written as a matrix. The only difference between "kernel of a linear transformation" and "kernel of a matrix" would be in the case of a linear transformation over infinite dimensional spaces which cannot be written as a matrix. An example would be the "differentiation" operator on the space of all differentiable functions. The kernel of that operator is the subspace of all constant functions.







                  share|cite|improve this answer












                  share|cite|improve this answer



                  share|cite|improve this answer










                  answered Dec 15 '18 at 15:32









                  user247327user247327

                  11.1k1515




                  11.1k1515






























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