How to define this application g to meet $Im space g subseteq T$?












1












$begingroup$


Let $S$ be the subspace of matrix 2x2 in $mathbb R$ formed by symetrical matrix and $T$ the one formed by the matrix of trace zero.



Let $A=left(begin{array}{cc} 1 & 0\ 2 & 0 end{array}right)$ and $f:Srightarrow T$ linear given by $f(M)=AM-MA.$



So the first question is to get $mathcal B_1$ and $mathcal B_2$ basis os $S$ and $T$ so the matrix of $f$ is $A=left(begin{array}{cc} I_r & 0\ 0 & 0 end{array}right)$.



So I ended up with:



$mathcal B_1$ :$left(begin{array}{cc} 0& 0\ 1 & 0 end{array}right)$$left(begin{array}{cc} 0 & 1\ 0 & 0 end{array}right)$$left(begin{array}{cc} 1 & 0\ 0 & 1 end{array}right)$.



$mathcal B_2$ : $left(begin{array}{cc} 0& 0\ -1 & 0 end{array}right)$$left(begin{array}{cc} -2 & 1\ 0 & 2 end{array}right)$$left(begin{array}{cc} 1 & 0\ 1 & 0 end{array}right)$



then the matrix of $f$ is: $left(begin{array}{cc} 1 & 0 & 0\ 0 & 1 &0\0&0&0end{array}right)$.



But now they ask me to find the applications $g:Srightarrow S$ with $fcirc g=f$ and express their matrix respect $mathcal B_1$(g is linear).



So here I got that those g are the ones with matrix: $left(begin{array}{cc} 1 & 0 & 0\ 0 & 1 &0\a&b&cend{array}right)$ .



With $a,b,cin mathbb R$.
But I don't know how to start with the next question.
Of all those g, find the ONLY ONE that meets $Imspace g subseteq T$.



How to start? Any hint?






linear-algebra algebra-precalculus







share|cite|improve this question











$endgroup$












  • $begingroup$
    How are the members of your $mathcal B_i$ elements of $S$ or $T$?
    $endgroup$
    – Berci
    Dec 31 '18 at 0:39


















1












$begingroup$


Let $S$ be the subspace of matrix 2x2 in $mathbb R$ formed by symetrical matrix and $T$ the one formed by the matrix of trace zero.



Let $A=left(begin{array}{cc} 1 & 0\ 2 & 0 end{array}right)$ and $f:Srightarrow T$ linear given by $f(M)=AM-MA.$



So the first question is to get $mathcal B_1$ and $mathcal B_2$ basis os $S$ and $T$ so the matrix of $f$ is $A=left(begin{array}{cc} I_r & 0\ 0 & 0 end{array}right)$.



So I ended up with:



$mathcal B_1$ :$left(begin{array}{cc} 0& 0\ 1 & 0 end{array}right)$$left(begin{array}{cc} 0 & 1\ 0 & 0 end{array}right)$$left(begin{array}{cc} 1 & 0\ 0 & 1 end{array}right)$.



$mathcal B_2$ : $left(begin{array}{cc} 0& 0\ -1 & 0 end{array}right)$$left(begin{array}{cc} -2 & 1\ 0 & 2 end{array}right)$$left(begin{array}{cc} 1 & 0\ 1 & 0 end{array}right)$



then the matrix of $f$ is: $left(begin{array}{cc} 1 & 0 & 0\ 0 & 1 &0\0&0&0end{array}right)$.



But now they ask me to find the applications $g:Srightarrow S$ with $fcirc g=f$ and express their matrix respect $mathcal B_1$(g is linear).



So here I got that those g are the ones with matrix: $left(begin{array}{cc} 1 & 0 & 0\ 0 & 1 &0\a&b&cend{array}right)$ .



With $a,b,cin mathbb R$.
But I don't know how to start with the next question.
Of all those g, find the ONLY ONE that meets $Imspace g subseteq T$.



How to start? Any hint?






linear-algebra algebra-precalculus







share|cite|improve this question











$endgroup$












  • $begingroup$
    How are the members of your $mathcal B_i$ elements of $S$ or $T$?
    $endgroup$
    – Berci
    Dec 31 '18 at 0:39
















1












1








1





$begingroup$


Let $S$ be the subspace of matrix 2x2 in $mathbb R$ formed by symetrical matrix and $T$ the one formed by the matrix of trace zero.



Let $A=left(begin{array}{cc} 1 & 0\ 2 & 0 end{array}right)$ and $f:Srightarrow T$ linear given by $f(M)=AM-MA.$



So the first question is to get $mathcal B_1$ and $mathcal B_2$ basis os $S$ and $T$ so the matrix of $f$ is $A=left(begin{array}{cc} I_r & 0\ 0 & 0 end{array}right)$.



So I ended up with:



$mathcal B_1$ :$left(begin{array}{cc} 0& 0\ 1 & 0 end{array}right)$$left(begin{array}{cc} 0 & 1\ 0 & 0 end{array}right)$$left(begin{array}{cc} 1 & 0\ 0 & 1 end{array}right)$.



$mathcal B_2$ : $left(begin{array}{cc} 0& 0\ -1 & 0 end{array}right)$$left(begin{array}{cc} -2 & 1\ 0 & 2 end{array}right)$$left(begin{array}{cc} 1 & 0\ 1 & 0 end{array}right)$



then the matrix of $f$ is: $left(begin{array}{cc} 1 & 0 & 0\ 0 & 1 &0\0&0&0end{array}right)$.



But now they ask me to find the applications $g:Srightarrow S$ with $fcirc g=f$ and express their matrix respect $mathcal B_1$(g is linear).



So here I got that those g are the ones with matrix: $left(begin{array}{cc} 1 & 0 & 0\ 0 & 1 &0\a&b&cend{array}right)$ .



With $a,b,cin mathbb R$.
But I don't know how to start with the next question.
Of all those g, find the ONLY ONE that meets $Imspace g subseteq T$.



How to start? Any hint?






linear-algebra algebra-precalculus







share|cite|improve this question











$endgroup$




Let $S$ be the subspace of matrix 2x2 in $mathbb R$ formed by symetrical matrix and $T$ the one formed by the matrix of trace zero.



Let $A=left(begin{array}{cc} 1 & 0\ 2 & 0 end{array}right)$ and $f:Srightarrow T$ linear given by $f(M)=AM-MA.$



So the first question is to get $mathcal B_1$ and $mathcal B_2$ basis os $S$ and $T$ so the matrix of $f$ is $A=left(begin{array}{cc} I_r & 0\ 0 & 0 end{array}right)$.



So I ended up with:



$mathcal B_1$ :$left(begin{array}{cc} 0& 0\ 1 & 0 end{array}right)$$left(begin{array}{cc} 0 & 1\ 0 & 0 end{array}right)$$left(begin{array}{cc} 1 & 0\ 0 & 1 end{array}right)$.



$mathcal B_2$ : $left(begin{array}{cc} 0& 0\ -1 & 0 end{array}right)$$left(begin{array}{cc} -2 & 1\ 0 & 2 end{array}right)$$left(begin{array}{cc} 1 & 0\ 1 & 0 end{array}right)$



then the matrix of $f$ is: $left(begin{array}{cc} 1 & 0 & 0\ 0 & 1 &0\0&0&0end{array}right)$.



But now they ask me to find the applications $g:Srightarrow S$ with $fcirc g=f$ and express their matrix respect $mathcal B_1$(g is linear).



So here I got that those g are the ones with matrix: $left(begin{array}{cc} 1 & 0 & 0\ 0 & 1 &0\a&b&cend{array}right)$ .



With $a,b,cin mathbb R$.
But I don't know how to start with the next question.
Of all those g, find the ONLY ONE that meets $Imspace g subseteq T$.



How to start? Any hint?






linear-algebra algebra-precalculus




linear-algebra algebra-precalculus






share|cite|improve this question















share|cite|improve this question













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edited Dec 31 '18 at 10:10







iggykimi

















asked Dec 31 '18 at 0:33









iggykimiiggykimi

31210




31210












  • $begingroup$
    How are the members of your $mathcal B_i$ elements of $S$ or $T$?
    $endgroup$
    – Berci
    Dec 31 '18 at 0:39




















  • $begingroup$
    How are the members of your $mathcal B_i$ elements of $S$ or $T$?
    $endgroup$
    – Berci
    Dec 31 '18 at 0:39


















$begingroup$
How are the members of your $mathcal B_i$ elements of $S$ or $T$?
$endgroup$
– Berci
Dec 31 '18 at 0:39






$begingroup$
How are the members of your $mathcal B_i$ elements of $S$ or $T$?
$endgroup$
– Berci
Dec 31 '18 at 0:39












1 Answer
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$begingroup$

Hints:




  • note that $Scap T = langleleft(begin{matrix}0 & 1 \ 1 & 0 end{matrix}right) rangle$

  • $fleft(begin{matrix}m_{11} & m_{12} \ m_{21}& m_{22} end{matrix}right)=left(begin{matrix} 0 & -m_{12} \ -m_{21}& 0 end{matrix}right)$






share|cite|improve this answer









$endgroup$













  • $begingroup$
    still don't know how to follow :(
    $endgroup$
    – iggykimi
    Dec 31 '18 at 14:49











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1












$begingroup$

Hints:




  • note that $Scap T = langleleft(begin{matrix}0 & 1 \ 1 & 0 end{matrix}right) rangle$

  • $fleft(begin{matrix}m_{11} & m_{12} \ m_{21}& m_{22} end{matrix}right)=left(begin{matrix} 0 & -m_{12} \ -m_{21}& 0 end{matrix}right)$






share|cite|improve this answer









$endgroup$













  • $begingroup$
    still don't know how to follow :(
    $endgroup$
    – iggykimi
    Dec 31 '18 at 14:49
















1












$begingroup$

Hints:




  • note that $Scap T = langleleft(begin{matrix}0 & 1 \ 1 & 0 end{matrix}right) rangle$

  • $fleft(begin{matrix}m_{11} & m_{12} \ m_{21}& m_{22} end{matrix}right)=left(begin{matrix} 0 & -m_{12} \ -m_{21}& 0 end{matrix}right)$






share|cite|improve this answer









$endgroup$













  • $begingroup$
    still don't know how to follow :(
    $endgroup$
    – iggykimi
    Dec 31 '18 at 14:49














1












1








1





$begingroup$

Hints:




  • note that $Scap T = langleleft(begin{matrix}0 & 1 \ 1 & 0 end{matrix}right) rangle$

  • $fleft(begin{matrix}m_{11} & m_{12} \ m_{21}& m_{22} end{matrix}right)=left(begin{matrix} 0 & -m_{12} \ -m_{21}& 0 end{matrix}right)$






share|cite|improve this answer









$endgroup$



Hints:




  • note that $Scap T = langleleft(begin{matrix}0 & 1 \ 1 & 0 end{matrix}right) rangle$

  • $fleft(begin{matrix}m_{11} & m_{12} \ m_{21}& m_{22} end{matrix}right)=left(begin{matrix} 0 & -m_{12} \ -m_{21}& 0 end{matrix}right)$







share|cite|improve this answer












share|cite|improve this answer



share|cite|improve this answer










answered Dec 31 '18 at 0:40









Martín Vacas VignoloMartín Vacas Vignolo

3,816623




3,816623












  • $begingroup$
    still don't know how to follow :(
    $endgroup$
    – iggykimi
    Dec 31 '18 at 14:49


















  • $begingroup$
    still don't know how to follow :(
    $endgroup$
    – iggykimi
    Dec 31 '18 at 14:49
















$begingroup$
still don't know how to follow :(
$endgroup$
– iggykimi
Dec 31 '18 at 14:49




$begingroup$
still don't know how to follow :(
$endgroup$
– iggykimi
Dec 31 '18 at 14:49


















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