find the dimension of $P_{100} $?











up vote
3
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let $P_n$ denote the real vector space of all polynomials in two variables
of degree strictly less than $n$ for $n ge 1, n in mathbb{N}$



find the dimension of $P_{100} $?



i thinks dimension of $P_{100} = 101$ beacause dimension of $P_{n}= n+1$



Is it correct ?






linear-algebra







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  • 2




    No. For instance, in $P_3,$ the polynomials $1,$ $x,$ $y,$ $xy$ $x^2$ and $y^2$ are a basis. Hence, $P_3$ has dimension 6.
    – Will M.
    Nov 22 at 20:25

















up vote
3
down vote

favorite












let $P_n$ denote the real vector space of all polynomials in two variables
of degree strictly less than $n$ for $n ge 1, n in mathbb{N}$



find the dimension of $P_{100} $?



i thinks dimension of $P_{100} = 101$ beacause dimension of $P_{n}= n+1$



Is it correct ?






linear-algebra







share|cite|improve this question


















  • 2




    No. For instance, in $P_3,$ the polynomials $1,$ $x,$ $y,$ $xy$ $x^2$ and $y^2$ are a basis. Hence, $P_3$ has dimension 6.
    – Will M.
    Nov 22 at 20:25















up vote
3
down vote

favorite









up vote
3
down vote

favorite











let $P_n$ denote the real vector space of all polynomials in two variables
of degree strictly less than $n$ for $n ge 1, n in mathbb{N}$



find the dimension of $P_{100} $?



i thinks dimension of $P_{100} = 101$ beacause dimension of $P_{n}= n+1$



Is it correct ?






linear-algebra







share|cite|improve this question













let $P_n$ denote the real vector space of all polynomials in two variables
of degree strictly less than $n$ for $n ge 1, n in mathbb{N}$



find the dimension of $P_{100} $?



i thinks dimension of $P_{100} = 101$ beacause dimension of $P_{n}= n+1$



Is it correct ?






linear-algebra




linear-algebra






share|cite|improve this question













share|cite|improve this question











share|cite|improve this question




share|cite|improve this question










asked Nov 22 at 20:23









Messi fifa

51611




51611








  • 2




    No. For instance, in $P_3,$ the polynomials $1,$ $x,$ $y,$ $xy$ $x^2$ and $y^2$ are a basis. Hence, $P_3$ has dimension 6.
    – Will M.
    Nov 22 at 20:25
















  • 2




    No. For instance, in $P_3,$ the polynomials $1,$ $x,$ $y,$ $xy$ $x^2$ and $y^2$ are a basis. Hence, $P_3$ has dimension 6.
    – Will M.
    Nov 22 at 20:25










2




2




No. For instance, in $P_3,$ the polynomials $1,$ $x,$ $y,$ $xy$ $x^2$ and $y^2$ are a basis. Hence, $P_3$ has dimension 6.
– Will M.
Nov 22 at 20:25






No. For instance, in $P_3,$ the polynomials $1,$ $x,$ $y,$ $xy$ $x^2$ and $y^2$ are a basis. Hence, $P_3$ has dimension 6.
– Will M.
Nov 22 at 20:25












1 Answer
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up vote
2
down vote



accepted










Let us consider the following:



$begin{align} &underline{text{Degree}}& underline{text{Number of basis vectors}}
\&0 &1quad {text{$1$}}\&1 &2 quad{text{$x,y$}} \&2 &3 quad{text{$x^2,xy,y^2$}}\&vdots &vdots\&99 &100quad{text{$x^{99},x^{98}y,cdots,y^{99}$}}end{align}$



So dimension of $P_{100}=dfrac{100times101}{2}=4050$.



(Here we have assumed that the variables $x$ and $y$ commute. As an exercise you may try to find $P_{100}$ when $x$ and $y$ do not commute.)






share|cite|improve this answer





















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    1 Answer
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    active

    oldest

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    1 Answer
    1






    active

    oldest

    votes









    active

    oldest

    votes






    active

    oldest

    votes








    up vote
    2
    down vote



    accepted










    Let us consider the following:



    $begin{align} &underline{text{Degree}}& underline{text{Number of basis vectors}}
    \&0 &1quad {text{$1$}}\&1 &2 quad{text{$x,y$}} \&2 &3 quad{text{$x^2,xy,y^2$}}\&vdots &vdots\&99 &100quad{text{$x^{99},x^{98}y,cdots,y^{99}$}}end{align}$



    So dimension of $P_{100}=dfrac{100times101}{2}=4050$.



    (Here we have assumed that the variables $x$ and $y$ commute. As an exercise you may try to find $P_{100}$ when $x$ and $y$ do not commute.)






    share|cite|improve this answer

























      up vote
      2
      down vote



      accepted










      Let us consider the following:



      $begin{align} &underline{text{Degree}}& underline{text{Number of basis vectors}}
      \&0 &1quad {text{$1$}}\&1 &2 quad{text{$x,y$}} \&2 &3 quad{text{$x^2,xy,y^2$}}\&vdots &vdots\&99 &100quad{text{$x^{99},x^{98}y,cdots,y^{99}$}}end{align}$



      So dimension of $P_{100}=dfrac{100times101}{2}=4050$.



      (Here we have assumed that the variables $x$ and $y$ commute. As an exercise you may try to find $P_{100}$ when $x$ and $y$ do not commute.)






      share|cite|improve this answer























        up vote
        2
        down vote



        accepted







        up vote
        2
        down vote



        accepted






        Let us consider the following:



        $begin{align} &underline{text{Degree}}& underline{text{Number of basis vectors}}
        \&0 &1quad {text{$1$}}\&1 &2 quad{text{$x,y$}} \&2 &3 quad{text{$x^2,xy,y^2$}}\&vdots &vdots\&99 &100quad{text{$x^{99},x^{98}y,cdots,y^{99}$}}end{align}$



        So dimension of $P_{100}=dfrac{100times101}{2}=4050$.



        (Here we have assumed that the variables $x$ and $y$ commute. As an exercise you may try to find $P_{100}$ when $x$ and $y$ do not commute.)






        share|cite|improve this answer












        Let us consider the following:



        $begin{align} &underline{text{Degree}}& underline{text{Number of basis vectors}}
        \&0 &1quad {text{$1$}}\&1 &2 quad{text{$x,y$}} \&2 &3 quad{text{$x^2,xy,y^2$}}\&vdots &vdots\&99 &100quad{text{$x^{99},x^{98}y,cdots,y^{99}$}}end{align}$



        So dimension of $P_{100}=dfrac{100times101}{2}=4050$.



        (Here we have assumed that the variables $x$ and $y$ commute. As an exercise you may try to find $P_{100}$ when $x$ and $y$ do not commute.)







        share|cite|improve this answer












        share|cite|improve this answer



        share|cite|improve this answer










        answered Nov 22 at 20:51









        Yadati Kiran

        1,354418




        1,354418






























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