Is set of all rational linear combination of function sin nx and cos mx $m,nin mathbb N cup {0}$?
$begingroup$
Is set of all rational linear combination of function sin nx and cos mx $m,nin mathbb N cup {0}$ is subring of ring of all function from $[0,1]to mathbb R $?
My attempt
$sin x:[0,2 pi]to[-1,1]$: is surjective function.
To show subring we have to show that closed under subtraction and multiplication.
Take $U_1=k_1 sin k_2x+k_3 cos k_4 x , U_2=sin k_6x+k_7 cos k_8x $
Now problem I incurred how to show $U_1-U_2in A$ and $U_1U_2in A$ where A is given subset
Please give me hint
abstract-algebra ring-theory
edited Jan 12 at 10:44
Fabio Lucchini
8,85811426
asked Jan 12 at 5:41
MathLoverMathLover
53710
$endgroup$
add a comment |
$begingroup$
Is set of all rational linear combination of function sin nx and cos mx $m,nin mathbb N cup {0}$ is subring of ring of all function from $[0,1]to mathbb R $?
My attempt
$sin x:[0,2 pi]to[-1,1]$: is surjective function.
To show subring we have to show that closed under subtraction and multiplication.
Take $U_1=k_1 sin k_2x+k_3 cos k_4 x , U_2=sin k_6x+k_7 cos k_8x $
Now problem I incurred how to show $U_1-U_2in A$ and $U_1U_2in A$ where A is given subset
Please give me hint
abstract-algebra ring-theory
edited Jan 12 at 10:44
Fabio Lucchini
8,85811426
asked Jan 12 at 5:41
MathLoverMathLover
53710
$endgroup$
$begingroup$
To produce ${$ you can use{, and similarly for $}$ you can use}.
$endgroup$
– Asaf Karagila♦
Jan 12 at 9:41
add a comment |
$begingroup$
Is set of all rational linear combination of function sin nx and cos mx $m,nin mathbb N cup {0}$ is subring of ring of all function from $[0,1]to mathbb R $?
My attempt
$sin x:[0,2 pi]to[-1,1]$: is surjective function.
To show subring we have to show that closed under subtraction and multiplication.
Take $U_1=k_1 sin k_2x+k_3 cos k_4 x , U_2=sin k_6x+k_7 cos k_8x $
Now problem I incurred how to show $U_1-U_2in A$ and $U_1U_2in A$ where A is given subset
Please give me hint
abstract-algebra ring-theory
edited Jan 12 at 10:44
Fabio Lucchini
8,85811426
asked Jan 12 at 5:41
MathLoverMathLover
53710
$endgroup$
Is set of all rational linear combination of function sin nx and cos mx $m,nin mathbb N cup {0}$ is subring of ring of all function from $[0,1]to mathbb R $?
My attempt
$sin x:[0,2 pi]to[-1,1]$: is surjective function.
To show subring we have to show that closed under subtraction and multiplication.
Take $U_1=k_1 sin k_2x+k_3 cos k_4 x , U_2=sin k_6x+k_7 cos k_8x $
Now problem I incurred how to show $U_1-U_2in A$ and $U_1U_2in A$ where A is given subset
Please give me hint
abstract-algebra ring-theory
abstract-algebra ring-theory
edited Jan 12 at 10:44
Fabio Lucchini
8,85811426
asked Jan 12 at 5:41
MathLoverMathLover
53710
edited Jan 12 at 10:44
Fabio Lucchini
8,85811426
asked Jan 12 at 5:41
MathLoverMathLover
53710
edited Jan 12 at 10:44
Fabio Lucchini
8,85811426
edited Jan 12 at 10:44
Fabio Lucchini
8,85811426
edited Jan 12 at 10:44
Fabio Lucchini
8,85811426
8,85811426
asked Jan 12 at 5:41
MathLoverMathLover
53710
asked Jan 12 at 5:41
MathLoverMathLover
53710
asked Jan 12 at 5:41
MathLoverMathLover
53710
53710
$begingroup$
To produce ${$ you can use{, and similarly for $}$ you can use}.
$endgroup$
– Asaf Karagila♦
Jan 12 at 9:41
add a comment |
$begingroup$
To produce ${$ you can use{, and similarly for $}$ you can use}.
$endgroup$
– Asaf Karagila♦
Jan 12 at 9:41
$begingroup$
To produce ${$ you can use
{, and similarly for $}$ you can use }.$endgroup$
– Asaf Karagila♦
Jan 12 at 9:41
$begingroup$
To produce ${$ you can use
{, and similarly for $}$ you can use }.$endgroup$
– Asaf Karagila♦
Jan 12 at 9:41
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
To get a ring, let $A $ the set of
$$sum_{i=1}^r (a_isin (n_ix)+b_icos (m_ix)) $$
with $a_i,b_iinBbb Q $ and $n_i,m_iinBbb N$.
Then $A $ is closed respect to sum.
Closuresness respect to product follows by Werner formulas.
answered Jan 12 at 8:20
Fabio LucchiniFabio Lucchini
8,85811426
$endgroup$
add a comment |
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1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
To get a ring, let $A $ the set of
$$sum_{i=1}^r (a_isin (n_ix)+b_icos (m_ix)) $$
with $a_i,b_iinBbb Q $ and $n_i,m_iinBbb N$.
Then $A $ is closed respect to sum.
Closuresness respect to product follows by Werner formulas.
answered Jan 12 at 8:20
Fabio LucchiniFabio Lucchini
8,85811426
$endgroup$
add a comment |
$begingroup$
To get a ring, let $A $ the set of
$$sum_{i=1}^r (a_isin (n_ix)+b_icos (m_ix)) $$
with $a_i,b_iinBbb Q $ and $n_i,m_iinBbb N$.
Then $A $ is closed respect to sum.
Closuresness respect to product follows by Werner formulas.
answered Jan 12 at 8:20
Fabio LucchiniFabio Lucchini
8,85811426
$endgroup$
add a comment |
$begingroup$
To get a ring, let $A $ the set of
$$sum_{i=1}^r (a_isin (n_ix)+b_icos (m_ix)) $$
with $a_i,b_iinBbb Q $ and $n_i,m_iinBbb N$.
Then $A $ is closed respect to sum.
Closuresness respect to product follows by Werner formulas.
answered Jan 12 at 8:20
Fabio LucchiniFabio Lucchini
8,85811426
$endgroup$
To get a ring, let $A $ the set of
$$sum_{i=1}^r (a_isin (n_ix)+b_icos (m_ix)) $$
with $a_i,b_iinBbb Q $ and $n_i,m_iinBbb N$.
Then $A $ is closed respect to sum.
Closuresness respect to product follows by Werner formulas.
answered Jan 12 at 8:20
Fabio LucchiniFabio Lucchini
8,85811426
edited Jan 12 at 10:45
answered Jan 12 at 8:20
Fabio LucchiniFabio Lucchini
8,85811426
answered Jan 12 at 8:20
Fabio LucchiniFabio Lucchini
8,85811426
answered Jan 12 at 8:20
Fabio LucchiniFabio Lucchini
8,85811426
8,85811426
add a comment |
add a comment |
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$begingroup$
To produce ${$ you can use
{, and similarly for $}$ you can use}.$endgroup$
– Asaf Karagila♦
Jan 12 at 9:41