Real n-by-n Matrices…

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5












$begingroup$


Let $M_n(mathbbR)$ denote the vector space of real $ntimes n$ matrics, and let $A in M_n(mathbbR)$.



Part (a) of this question says: Suppose $B in M_n(mathbbR)$ such that $AB = I_n$ (the $n times n$ identity matrix. If $C in M_n(mathbbR)$ such that $CA = 0$, then prove $C = 0$.



I have already proven part (a).



Part (b) asks: Assume there exists a least positive integer $m$ such that $t_0I + t_1A + dots + t_mA^m = 0$ for some $t_0, dots, t_m in mathbbR$ with $t_m neq 0$. Also, suppose that $AB = I_n$ for some $B in M_n(mathbbR)$.Prove that $t_0 neq 0$. (Hint: Use the result from part (a)).



My idea is to use induction on $m$. That is, suppose



$$t_0I = 0.$$



But this implies that $t_0 = 0$ since $I$ is the identity. But we could see this as



beginalign*
t_0I &= 0
\
t_0AB &= 0
\
t_0CAB &= 0
\
t_0C &= 0.
endalign*



But I don't think this is the right approach. So I have a few questions: (i) Is this the correct approach? (ii) How do I use the result from part (a) properly? (iii) What's a good resource for these types of questions? The book that I am using is "Linear Algebra Done Right by Sheldon Axler".



I am studying for my linear algebra comp in a few weeks so any help is appreciated!










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  • $begingroup$
    Once you have got $t_0=0$ then anything you multiply it with gonna be zero. Since we assume $t_mne 0$ then $mge 1$.
    $endgroup$
    – A.Γ.
    Jan 1 at 20:44















5












$begingroup$


Let $M_n(mathbbR)$ denote the vector space of real $ntimes n$ matrics, and let $A in M_n(mathbbR)$.



Part (a) of this question says: Suppose $B in M_n(mathbbR)$ such that $AB = I_n$ (the $n times n$ identity matrix. If $C in M_n(mathbbR)$ such that $CA = 0$, then prove $C = 0$.



I have already proven part (a).



Part (b) asks: Assume there exists a least positive integer $m$ such that $t_0I + t_1A + dots + t_mA^m = 0$ for some $t_0, dots, t_m in mathbbR$ with $t_m neq 0$. Also, suppose that $AB = I_n$ for some $B in M_n(mathbbR)$.Prove that $t_0 neq 0$. (Hint: Use the result from part (a)).



My idea is to use induction on $m$. That is, suppose



$$t_0I = 0.$$



But this implies that $t_0 = 0$ since $I$ is the identity. But we could see this as



beginalign*
t_0I &= 0
\
t_0AB &= 0
\
t_0CAB &= 0
\
t_0C &= 0.
endalign*



But I don't think this is the right approach. So I have a few questions: (i) Is this the correct approach? (ii) How do I use the result from part (a) properly? (iii) What's a good resource for these types of questions? The book that I am using is "Linear Algebra Done Right by Sheldon Axler".



I am studying for my linear algebra comp in a few weeks so any help is appreciated!










share|cite|improve this question









$endgroup$











  • $begingroup$
    Once you have got $t_0=0$ then anything you multiply it with gonna be zero. Since we assume $t_mne 0$ then $mge 1$.
    $endgroup$
    – A.Γ.
    Jan 1 at 20:44













5












5








5





$begingroup$


Let $M_n(mathbbR)$ denote the vector space of real $ntimes n$ matrics, and let $A in M_n(mathbbR)$.



Part (a) of this question says: Suppose $B in M_n(mathbbR)$ such that $AB = I_n$ (the $n times n$ identity matrix. If $C in M_n(mathbbR)$ such that $CA = 0$, then prove $C = 0$.



I have already proven part (a).



Part (b) asks: Assume there exists a least positive integer $m$ such that $t_0I + t_1A + dots + t_mA^m = 0$ for some $t_0, dots, t_m in mathbbR$ with $t_m neq 0$. Also, suppose that $AB = I_n$ for some $B in M_n(mathbbR)$.Prove that $t_0 neq 0$. (Hint: Use the result from part (a)).



My idea is to use induction on $m$. That is, suppose



$$t_0I = 0.$$



But this implies that $t_0 = 0$ since $I$ is the identity. But we could see this as



beginalign*
t_0I &= 0
\
t_0AB &= 0
\
t_0CAB &= 0
\
t_0C &= 0.
endalign*



But I don't think this is the right approach. So I have a few questions: (i) Is this the correct approach? (ii) How do I use the result from part (a) properly? (iii) What's a good resource for these types of questions? The book that I am using is "Linear Algebra Done Right by Sheldon Axler".



I am studying for my linear algebra comp in a few weeks so any help is appreciated!










share|cite|improve this question









$endgroup$




Let $M_n(mathbbR)$ denote the vector space of real $ntimes n$ matrics, and let $A in M_n(mathbbR)$.



Part (a) of this question says: Suppose $B in M_n(mathbbR)$ such that $AB = I_n$ (the $n times n$ identity matrix. If $C in M_n(mathbbR)$ such that $CA = 0$, then prove $C = 0$.



I have already proven part (a).



Part (b) asks: Assume there exists a least positive integer $m$ such that $t_0I + t_1A + dots + t_mA^m = 0$ for some $t_0, dots, t_m in mathbbR$ with $t_m neq 0$. Also, suppose that $AB = I_n$ for some $B in M_n(mathbbR)$.Prove that $t_0 neq 0$. (Hint: Use the result from part (a)).



My idea is to use induction on $m$. That is, suppose



$$t_0I = 0.$$



But this implies that $t_0 = 0$ since $I$ is the identity. But we could see this as



beginalign*
t_0I &= 0
\
t_0AB &= 0
\
t_0CAB &= 0
\
t_0C &= 0.
endalign*



But I don't think this is the right approach. So I have a few questions: (i) Is this the correct approach? (ii) How do I use the result from part (a) properly? (iii) What's a good resource for these types of questions? The book that I am using is "Linear Algebra Done Right by Sheldon Axler".



I am studying for my linear algebra comp in a few weeks so any help is appreciated!







linear-algebra matrices vector-spaces






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asked Jan 1 at 20:25









Taylor McMillanTaylor McMillan

695




695











  • $begingroup$
    Once you have got $t_0=0$ then anything you multiply it with gonna be zero. Since we assume $t_mne 0$ then $mge 1$.
    $endgroup$
    – A.Γ.
    Jan 1 at 20:44
















  • $begingroup$
    Once you have got $t_0=0$ then anything you multiply it with gonna be zero. Since we assume $t_mne 0$ then $mge 1$.
    $endgroup$
    – A.Γ.
    Jan 1 at 20:44















$begingroup$
Once you have got $t_0=0$ then anything you multiply it with gonna be zero. Since we assume $t_mne 0$ then $mge 1$.
$endgroup$
– A.Γ.
Jan 1 at 20:44




$begingroup$
Once you have got $t_0=0$ then anything you multiply it with gonna be zero. Since we assume $t_mne 0$ then $mge 1$.
$endgroup$
– A.Γ.
Jan 1 at 20:44










3 Answers
3






active

oldest

votes


















7












$begingroup$

Suppose exists $B$ such that $AB=I$ and that $m$ is the smallest integer such that, exist coefficients $t_i$ such that,
$$
t_0 I + t_1 A + cdots + t_m A^m = 0
$$



If $t_0=0$ you have,
$$
t_1 A + cdots + t_m A^m = 0
$$



Now, see if you can use the fact that $AB = I$ to find new coefficients $t_i'$ satisfying
$$
t_0'I + t_1'A + cdots t_n ' A^n = 0
$$

where $n<m$.



Since by assumption $m$ is the smallest integer such that this type of combination exists, then you have a contradiction.



I liked that book as a first course in linear algebra. You could also see if past exam problems are available to study from. To me, studying for exams is somewhat dependent on the format of exam itself so its hard to give general advice.






share|cite|improve this answer









$endgroup$












  • $begingroup$
    Thank you for this. My goto for using contradiction is typically when the problem screams "this is obviously true". Didn't really think that this result was intuitive so I didn't take this route. When in doubt try it out, I guess.
    $endgroup$
    – Taylor McMillan
    Jan 1 at 21:05










  • $begingroup$
    That is what I am doing. My university has several past exams posted I am doing all of those problems.
    $endgroup$
    – Taylor McMillan
    Jan 1 at 21:07


















4












$begingroup$

Hint: assume $t_0=0$ then
$$
t_1A+ldots+t_mA^m=0.
$$

Multiply by $B$ and get the contradiction with the minimality of $m$.






share|cite|improve this answer









$endgroup$




















    1












    $begingroup$

    Here is another hint/solution:



    • assume it to equal $0$,

    • multiply your equation by $A$, and factor by $A$,

    • using part (a), get a contradiction.





    share|cite|improve this answer











    $endgroup$












    • $begingroup$
      Bullet lists need a blank line before them in order to appear—I've added one. Welcome to the site!
      $endgroup$
      – timtfj
      Jan 1 at 22:18










    Your Answer





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    3 Answers
    3






    active

    oldest

    votes








    3 Answers
    3






    active

    oldest

    votes









    active

    oldest

    votes






    active

    oldest

    votes









    7












    $begingroup$

    Suppose exists $B$ such that $AB=I$ and that $m$ is the smallest integer such that, exist coefficients $t_i$ such that,
    $$
    t_0 I + t_1 A + cdots + t_m A^m = 0
    $$



    If $t_0=0$ you have,
    $$
    t_1 A + cdots + t_m A^m = 0
    $$



    Now, see if you can use the fact that $AB = I$ to find new coefficients $t_i'$ satisfying
    $$
    t_0'I + t_1'A + cdots t_n ' A^n = 0
    $$

    where $n<m$.



    Since by assumption $m$ is the smallest integer such that this type of combination exists, then you have a contradiction.



    I liked that book as a first course in linear algebra. You could also see if past exam problems are available to study from. To me, studying for exams is somewhat dependent on the format of exam itself so its hard to give general advice.






    share|cite|improve this answer









    $endgroup$












    • $begingroup$
      Thank you for this. My goto for using contradiction is typically when the problem screams "this is obviously true". Didn't really think that this result was intuitive so I didn't take this route. When in doubt try it out, I guess.
      $endgroup$
      – Taylor McMillan
      Jan 1 at 21:05










    • $begingroup$
      That is what I am doing. My university has several past exams posted I am doing all of those problems.
      $endgroup$
      – Taylor McMillan
      Jan 1 at 21:07















    7












    $begingroup$

    Suppose exists $B$ such that $AB=I$ and that $m$ is the smallest integer such that, exist coefficients $t_i$ such that,
    $$
    t_0 I + t_1 A + cdots + t_m A^m = 0
    $$



    If $t_0=0$ you have,
    $$
    t_1 A + cdots + t_m A^m = 0
    $$



    Now, see if you can use the fact that $AB = I$ to find new coefficients $t_i'$ satisfying
    $$
    t_0'I + t_1'A + cdots t_n ' A^n = 0
    $$

    where $n<m$.



    Since by assumption $m$ is the smallest integer such that this type of combination exists, then you have a contradiction.



    I liked that book as a first course in linear algebra. You could also see if past exam problems are available to study from. To me, studying for exams is somewhat dependent on the format of exam itself so its hard to give general advice.






    share|cite|improve this answer









    $endgroup$












    • $begingroup$
      Thank you for this. My goto for using contradiction is typically when the problem screams "this is obviously true". Didn't really think that this result was intuitive so I didn't take this route. When in doubt try it out, I guess.
      $endgroup$
      – Taylor McMillan
      Jan 1 at 21:05










    • $begingroup$
      That is what I am doing. My university has several past exams posted I am doing all of those problems.
      $endgroup$
      – Taylor McMillan
      Jan 1 at 21:07













    7












    7








    7





    $begingroup$

    Suppose exists $B$ such that $AB=I$ and that $m$ is the smallest integer such that, exist coefficients $t_i$ such that,
    $$
    t_0 I + t_1 A + cdots + t_m A^m = 0
    $$



    If $t_0=0$ you have,
    $$
    t_1 A + cdots + t_m A^m = 0
    $$



    Now, see if you can use the fact that $AB = I$ to find new coefficients $t_i'$ satisfying
    $$
    t_0'I + t_1'A + cdots t_n ' A^n = 0
    $$

    where $n<m$.



    Since by assumption $m$ is the smallest integer such that this type of combination exists, then you have a contradiction.



    I liked that book as a first course in linear algebra. You could also see if past exam problems are available to study from. To me, studying for exams is somewhat dependent on the format of exam itself so its hard to give general advice.






    share|cite|improve this answer









    $endgroup$



    Suppose exists $B$ such that $AB=I$ and that $m$ is the smallest integer such that, exist coefficients $t_i$ such that,
    $$
    t_0 I + t_1 A + cdots + t_m A^m = 0
    $$



    If $t_0=0$ you have,
    $$
    t_1 A + cdots + t_m A^m = 0
    $$



    Now, see if you can use the fact that $AB = I$ to find new coefficients $t_i'$ satisfying
    $$
    t_0'I + t_1'A + cdots t_n ' A^n = 0
    $$

    where $n<m$.



    Since by assumption $m$ is the smallest integer such that this type of combination exists, then you have a contradiction.



    I liked that book as a first course in linear algebra. You could also see if past exam problems are available to study from. To me, studying for exams is somewhat dependent on the format of exam itself so its hard to give general advice.







    share|cite|improve this answer












    share|cite|improve this answer



    share|cite|improve this answer










    answered Jan 1 at 20:46









    tchtch

    639210




    639210











    • $begingroup$
      Thank you for this. My goto for using contradiction is typically when the problem screams "this is obviously true". Didn't really think that this result was intuitive so I didn't take this route. When in doubt try it out, I guess.
      $endgroup$
      – Taylor McMillan
      Jan 1 at 21:05










    • $begingroup$
      That is what I am doing. My university has several past exams posted I am doing all of those problems.
      $endgroup$
      – Taylor McMillan
      Jan 1 at 21:07
















    • $begingroup$
      Thank you for this. My goto for using contradiction is typically when the problem screams "this is obviously true". Didn't really think that this result was intuitive so I didn't take this route. When in doubt try it out, I guess.
      $endgroup$
      – Taylor McMillan
      Jan 1 at 21:05










    • $begingroup$
      That is what I am doing. My university has several past exams posted I am doing all of those problems.
      $endgroup$
      – Taylor McMillan
      Jan 1 at 21:07















    $begingroup$
    Thank you for this. My goto for using contradiction is typically when the problem screams "this is obviously true". Didn't really think that this result was intuitive so I didn't take this route. When in doubt try it out, I guess.
    $endgroup$
    – Taylor McMillan
    Jan 1 at 21:05




    $begingroup$
    Thank you for this. My goto for using contradiction is typically when the problem screams "this is obviously true". Didn't really think that this result was intuitive so I didn't take this route. When in doubt try it out, I guess.
    $endgroup$
    – Taylor McMillan
    Jan 1 at 21:05












    $begingroup$
    That is what I am doing. My university has several past exams posted I am doing all of those problems.
    $endgroup$
    – Taylor McMillan
    Jan 1 at 21:07




    $begingroup$
    That is what I am doing. My university has several past exams posted I am doing all of those problems.
    $endgroup$
    – Taylor McMillan
    Jan 1 at 21:07











    4












    $begingroup$

    Hint: assume $t_0=0$ then
    $$
    t_1A+ldots+t_mA^m=0.
    $$

    Multiply by $B$ and get the contradiction with the minimality of $m$.






    share|cite|improve this answer









    $endgroup$

















      4












      $begingroup$

      Hint: assume $t_0=0$ then
      $$
      t_1A+ldots+t_mA^m=0.
      $$

      Multiply by $B$ and get the contradiction with the minimality of $m$.






      share|cite|improve this answer









      $endgroup$















        4












        4








        4





        $begingroup$

        Hint: assume $t_0=0$ then
        $$
        t_1A+ldots+t_mA^m=0.
        $$

        Multiply by $B$ and get the contradiction with the minimality of $m$.






        share|cite|improve this answer









        $endgroup$



        Hint: assume $t_0=0$ then
        $$
        t_1A+ldots+t_mA^m=0.
        $$

        Multiply by $B$ and get the contradiction with the minimality of $m$.







        share|cite|improve this answer












        share|cite|improve this answer



        share|cite|improve this answer










        answered Jan 1 at 20:46









        A.Γ.A.Γ.

        22.7k32656




        22.7k32656





















            1












            $begingroup$

            Here is another hint/solution:



            • assume it to equal $0$,

            • multiply your equation by $A$, and factor by $A$,

            • using part (a), get a contradiction.





            share|cite|improve this answer











            $endgroup$












            • $begingroup$
              Bullet lists need a blank line before them in order to appear—I've added one. Welcome to the site!
              $endgroup$
              – timtfj
              Jan 1 at 22:18















            1












            $begingroup$

            Here is another hint/solution:



            • assume it to equal $0$,

            • multiply your equation by $A$, and factor by $A$,

            • using part (a), get a contradiction.





            share|cite|improve this answer











            $endgroup$












            • $begingroup$
              Bullet lists need a blank line before them in order to appear—I've added one. Welcome to the site!
              $endgroup$
              – timtfj
              Jan 1 at 22:18













            1












            1








            1





            $begingroup$

            Here is another hint/solution:



            • assume it to equal $0$,

            • multiply your equation by $A$, and factor by $A$,

            • using part (a), get a contradiction.





            share|cite|improve this answer











            $endgroup$



            Here is another hint/solution:



            • assume it to equal $0$,

            • multiply your equation by $A$, and factor by $A$,

            • using part (a), get a contradiction.






            share|cite|improve this answer














            share|cite|improve this answer



            share|cite|improve this answer








            edited Jan 1 at 22:14









            amWhy

            192k28225439




            192k28225439










            answered Jan 1 at 21:25









            Cauchy is my masterCauchy is my master

            92




            92











            • $begingroup$
              Bullet lists need a blank line before them in order to appear—I've added one. Welcome to the site!
              $endgroup$
              – timtfj
              Jan 1 at 22:18
















            • $begingroup$
              Bullet lists need a blank line before them in order to appear—I've added one. Welcome to the site!
              $endgroup$
              – timtfj
              Jan 1 at 22:18















            $begingroup$
            Bullet lists need a blank line before them in order to appear—I've added one. Welcome to the site!
            $endgroup$
            – timtfj
            Jan 1 at 22:18




            $begingroup$
            Bullet lists need a blank line before them in order to appear—I've added one. Welcome to the site!
            $endgroup$
            – timtfj
            Jan 1 at 22:18

















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