find the dimension of $P_{100} $?
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
asked Nov 22 at 20:23
Messi fifa
50111
add a comment |
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
asked Nov 22 at 20:23
Messi fifa
50111
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
add a comment |
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
asked Nov 22 at 20:23
Messi fifa
50111
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
asked Nov 22 at 20:23
Messi fifa
50111
asked Nov 22 at 20:23
Messi fifa
50111
asked Nov 22 at 20:23
Messi fifa
50111
asked Nov 22 at 20:23
Messi fifa
50111
asked Nov 22 at 20:23
Messi fifa
50111
50111
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
add a comment |
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
add a comment |
1 Answer
1
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up vote
2
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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.)
answered Nov 22 at 20:51
Yadati Kiran
1,243417
add a comment |
1 Answer
1
active
oldest
votes
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.)
answered Nov 22 at 20:51
Yadati Kiran
1,243417
add a comment |
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.)
answered Nov 22 at 20:51
Yadati Kiran
1,243417
add a comment |
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.)
answered Nov 22 at 20:51
Yadati Kiran
1,243417
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.)
answered Nov 22 at 20:51
Yadati Kiran
1,243417
answered Nov 22 at 20:51
Yadati Kiran
1,243417
answered Nov 22 at 20:51
Yadati Kiran
1,243417
answered Nov 22 at 20:51
Yadati Kiran
1,243417
1,243417
add a comment |
add a comment |
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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