Linear maps even

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Let $V$ be an $n$-dimensional vector space.



To prove:
$DeclareMathOperatorimim$
There is a linear map $varphi:Vto V$ with $ker varphi= im
varphi $ if and only if $n$ is even.




Solution:



"$Rightarrow$" If $ker varphi=im varphi $ then $n = dim ker varphi+ dim im varphi = 2cdot dim ker varphi = 2cdot dim im varphi$



So $n$ is even.



How do I show the other direction "$Leftarrow$"?



Can I say if $n$ is even, we know from "$Rightarrow$" that there is a linear map?




linear-algebra




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    Let $V$ be an $n$-dimensional vector space.



    To prove:
    $DeclareMathOperatorimim$
    There is a linear map $varphi:Vto V$ with $ker varphi= im
    varphi $ if and only if $n$ is even.




    Solution:



    "$Rightarrow$" If $ker varphi=im varphi $ then $n = dim ker varphi+ dim im varphi = 2cdot dim ker varphi = 2cdot dim im varphi$



    So $n$ is even.



    How do I show the other direction "$Leftarrow$"?



    Can I say if $n$ is even, we know from "$Rightarrow$" that there is a linear map?




    linear-algebra




    share|cite|improve this question























      up vote
      1
      down vote

      favorite









      up vote
      1
      down vote

      favorite












      Let $V$ be an $n$-dimensional vector space.



      To prove:
      $DeclareMathOperatorimim$
      There is a linear map $varphi:Vto V$ with $ker varphi= im
      varphi $ if and only if $n$ is even.




      Solution:



      "$Rightarrow$" If $ker varphi=im varphi $ then $n = dim ker varphi+ dim im varphi = 2cdot dim ker varphi = 2cdot dim im varphi$



      So $n$ is even.



      How do I show the other direction "$Leftarrow$"?



      Can I say if $n$ is even, we know from "$Rightarrow$" that there is a linear map?




      linear-algebra




      share|cite|improve this question














      Let $V$ be an $n$-dimensional vector space.



      To prove:
      $DeclareMathOperatorimim$
      There is a linear map $varphi:Vto V$ with $ker varphi= im
      varphi $ if and only if $n$ is even.




      Solution:



      "$Rightarrow$" If $ker varphi=im varphi $ then $n = dim ker varphi+ dim im varphi = 2cdot dim ker varphi = 2cdot dim im varphi$



      So $n$ is even.



      How do I show the other direction "$Leftarrow$"?



      Can I say if $n$ is even, we know from "$Rightarrow$" that there is a linear map?





      linear-algebra





      share|cite|improve this question












      share|cite|improve this question




      share|cite|improve this question








      edited Jul 26 at 21:09









      Antoine

      2,485925




      2,485925









      asked Jul 26 at 20:56









      Marc

      1005




      1005




















          2 Answers
          2






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          up vote
          7
          down vote













          Select a basis $e_i_i=1^2N$. Map $e_i_i=1^N$ to 0, and $e_i_i=N+1^2N$ to $e_i_i=1^N$ (by taking $e_i$ to $e_-N+i $). This is a map with the desired property.






          share|cite|improve this answer





















          • That would be $varphi = (0_n,dotsc, 0_n, e_1, dotsc e_N)$. The first half mapped to zero the second half to a set of $N$ linear independent vectors.
            – mvw
            Jul 26 at 21:11











          • Yes, and the matrix in the answer below represents this transformation.
            – ertl
            Jul 26 at 21:13


















          up vote
          0
          down vote














          Can I say if $n$ is even, we know from "$Rightarrow$" that there is a linear map?




          Note: You want to infer from the even dimension, and $V$ being a vector space, that there exists a linear map $varphi$ from $V$ to $V$ with $ker phi = im phi$.






          share|cite|improve this answer























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            2 Answers
            2






            active

            oldest

            votes








            2 Answers
            2






            active

            oldest

            votes









            active

            oldest

            votes






            active

            oldest

            votes








            up vote
            7
            down vote













            Select a basis $e_i_i=1^2N$. Map $e_i_i=1^N$ to 0, and $e_i_i=N+1^2N$ to $e_i_i=1^N$ (by taking $e_i$ to $e_-N+i $). This is a map with the desired property.






            share|cite|improve this answer





















            • That would be $varphi = (0_n,dotsc, 0_n, e_1, dotsc e_N)$. The first half mapped to zero the second half to a set of $N$ linear independent vectors.
              – mvw
              Jul 26 at 21:11











            • Yes, and the matrix in the answer below represents this transformation.
              – ertl
              Jul 26 at 21:13















            up vote
            7
            down vote













            Select a basis $e_i_i=1^2N$. Map $e_i_i=1^N$ to 0, and $e_i_i=N+1^2N$ to $e_i_i=1^N$ (by taking $e_i$ to $e_-N+i $). This is a map with the desired property.






            share|cite|improve this answer





















            • That would be $varphi = (0_n,dotsc, 0_n, e_1, dotsc e_N)$. The first half mapped to zero the second half to a set of $N$ linear independent vectors.
              – mvw
              Jul 26 at 21:11











            • Yes, and the matrix in the answer below represents this transformation.
              – ertl
              Jul 26 at 21:13













            up vote
            7
            down vote










            up vote
            7
            down vote









            Select a basis $e_i_i=1^2N$. Map $e_i_i=1^N$ to 0, and $e_i_i=N+1^2N$ to $e_i_i=1^N$ (by taking $e_i$ to $e_-N+i $). This is a map with the desired property.






            share|cite|improve this answer













            Select a basis $e_i_i=1^2N$. Map $e_i_i=1^N$ to 0, and $e_i_i=N+1^2N$ to $e_i_i=1^N$ (by taking $e_i$ to $e_-N+i $). This is a map with the desired property.







            share|cite|improve this answer













            share|cite|improve this answer



            share|cite|improve this answer











            answered Jul 26 at 21:02









            ertl

            445110




            445110











            • That would be $varphi = (0_n,dotsc, 0_n, e_1, dotsc e_N)$. The first half mapped to zero the second half to a set of $N$ linear independent vectors.
              – mvw
              Jul 26 at 21:11











            • Yes, and the matrix in the answer below represents this transformation.
              – ertl
              Jul 26 at 21:13

















            • That would be $varphi = (0_n,dotsc, 0_n, e_1, dotsc e_N)$. The first half mapped to zero the second half to a set of $N$ linear independent vectors.
              – mvw
              Jul 26 at 21:11











            • Yes, and the matrix in the answer below represents this transformation.
              – ertl
              Jul 26 at 21:13
















            That would be $varphi = (0_n,dotsc, 0_n, e_1, dotsc e_N)$. The first half mapped to zero the second half to a set of $N$ linear independent vectors.
            – mvw
            Jul 26 at 21:11





            That would be $varphi = (0_n,dotsc, 0_n, e_1, dotsc e_N)$. The first half mapped to zero the second half to a set of $N$ linear independent vectors.
            – mvw
            Jul 26 at 21:11













            Yes, and the matrix in the answer below represents this transformation.
            – ertl
            Jul 26 at 21:13





            Yes, and the matrix in the answer below represents this transformation.
            – ertl
            Jul 26 at 21:13











            up vote
            0
            down vote














            Can I say if $n$ is even, we know from "$Rightarrow$" that there is a linear map?




            Note: You want to infer from the even dimension, and $V$ being a vector space, that there exists a linear map $varphi$ from $V$ to $V$ with $ker phi = im phi$.






            share|cite|improve this answer



























              up vote
              0
              down vote














              Can I say if $n$ is even, we know from "$Rightarrow$" that there is a linear map?




              Note: You want to infer from the even dimension, and $V$ being a vector space, that there exists a linear map $varphi$ from $V$ to $V$ with $ker phi = im phi$.






              share|cite|improve this answer

























                up vote
                0
                down vote










                up vote
                0
                down vote










                Can I say if $n$ is even, we know from "$Rightarrow$" that there is a linear map?




                Note: You want to infer from the even dimension, and $V$ being a vector space, that there exists a linear map $varphi$ from $V$ to $V$ with $ker phi = im phi$.






                share|cite|improve this answer
















                Can I say if $n$ is even, we know from "$Rightarrow$" that there is a linear map?




                Note: You want to infer from the even dimension, and $V$ being a vector space, that there exists a linear map $varphi$ from $V$ to $V$ with $ker phi = im phi$.







                share|cite|improve this answer















                share|cite|improve this answer



                share|cite|improve this answer








                edited Jul 26 at 21:15


























                answered Jul 26 at 21:06









                mvw

                30.2k22250




                30.2k22250






















                     

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