Why the set S(t) is the subspace of the null space of A?











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The picture below is on the page 5 of the paper On linear differential-algebraic equations and linearizations.



pic from the paper



What I have learnt from the pic is as follows:





  1. $N(t):=ker A(t)subset mathbb{R}^m$ is that $N(t)$ is the solution space of $A(t)x=0$.


  2. $B(t)z in mathbb{im} : A(t)$ is that $forall z,exists y, B(t)z=A(t)y$


So why we can get $S(t)$ is the subspace of the solution space of $A(t)x'(t)+B(t)x(t)=0, : tin J$



And what book do I need read next?



Thanks










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  • What they mean by "$S(t)$ is the subspace where the homogeneous solutions proceed" is that for the homogeneous equation, $x(t)$ has to lie in $S(t)$. This is because otherwise it would not be possible to satisfy the homogeneous equation for any value of $x'(t)$.
    – Rahul
    Nov 16 at 13:26












  • @Rahul Sorry, i don't know,That "$S(t)$ is the subspace of the homogeneous equation solutions" is that $forall z in S(t)$ is the solution of $A(t)x'(t)+B(t)x(t)=0$?? And $A(t)z'+B(t)z=0, B(t)z=A(t)y$
    – Hewie Ding
    Nov 16 at 14:07

















up vote
0
down vote

favorite












The picture below is on the page 5 of the paper On linear differential-algebraic equations and linearizations.



pic from the paper



What I have learnt from the pic is as follows:





  1. $N(t):=ker A(t)subset mathbb{R}^m$ is that $N(t)$ is the solution space of $A(t)x=0$.


  2. $B(t)z in mathbb{im} : A(t)$ is that $forall z,exists y, B(t)z=A(t)y$


So why we can get $S(t)$ is the subspace of the solution space of $A(t)x'(t)+B(t)x(t)=0, : tin J$



And what book do I need read next?



Thanks










share|cite|improve this question
























  • What they mean by "$S(t)$ is the subspace where the homogeneous solutions proceed" is that for the homogeneous equation, $x(t)$ has to lie in $S(t)$. This is because otherwise it would not be possible to satisfy the homogeneous equation for any value of $x'(t)$.
    – Rahul
    Nov 16 at 13:26












  • @Rahul Sorry, i don't know,That "$S(t)$ is the subspace of the homogeneous equation solutions" is that $forall z in S(t)$ is the solution of $A(t)x'(t)+B(t)x(t)=0$?? And $A(t)z'+B(t)z=0, B(t)z=A(t)y$
    – Hewie Ding
    Nov 16 at 14:07















up vote
0
down vote

favorite









up vote
0
down vote

favorite











The picture below is on the page 5 of the paper On linear differential-algebraic equations and linearizations.



pic from the paper



What I have learnt from the pic is as follows:





  1. $N(t):=ker A(t)subset mathbb{R}^m$ is that $N(t)$ is the solution space of $A(t)x=0$.


  2. $B(t)z in mathbb{im} : A(t)$ is that $forall z,exists y, B(t)z=A(t)y$


So why we can get $S(t)$ is the subspace of the solution space of $A(t)x'(t)+B(t)x(t)=0, : tin J$



And what book do I need read next?



Thanks










share|cite|improve this question















The picture below is on the page 5 of the paper On linear differential-algebraic equations and linearizations.



pic from the paper



What I have learnt from the pic is as follows:





  1. $N(t):=ker A(t)subset mathbb{R}^m$ is that $N(t)$ is the solution space of $A(t)x=0$.


  2. $B(t)z in mathbb{im} : A(t)$ is that $forall z,exists y, B(t)z=A(t)y$


So why we can get $S(t)$ is the subspace of the solution space of $A(t)x'(t)+B(t)x(t)=0, : tin J$



And what book do I need read next?



Thanks







linear-algebra differential-equations






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edited Nov 16 at 13:21









Rahul

32.9k466161




32.9k466161










asked Nov 16 at 12:52









Hewie Ding

13




13












  • What they mean by "$S(t)$ is the subspace where the homogeneous solutions proceed" is that for the homogeneous equation, $x(t)$ has to lie in $S(t)$. This is because otherwise it would not be possible to satisfy the homogeneous equation for any value of $x'(t)$.
    – Rahul
    Nov 16 at 13:26












  • @Rahul Sorry, i don't know,That "$S(t)$ is the subspace of the homogeneous equation solutions" is that $forall z in S(t)$ is the solution of $A(t)x'(t)+B(t)x(t)=0$?? And $A(t)z'+B(t)z=0, B(t)z=A(t)y$
    – Hewie Ding
    Nov 16 at 14:07




















  • What they mean by "$S(t)$ is the subspace where the homogeneous solutions proceed" is that for the homogeneous equation, $x(t)$ has to lie in $S(t)$. This is because otherwise it would not be possible to satisfy the homogeneous equation for any value of $x'(t)$.
    – Rahul
    Nov 16 at 13:26












  • @Rahul Sorry, i don't know,That "$S(t)$ is the subspace of the homogeneous equation solutions" is that $forall z in S(t)$ is the solution of $A(t)x'(t)+B(t)x(t)=0$?? And $A(t)z'+B(t)z=0, B(t)z=A(t)y$
    – Hewie Ding
    Nov 16 at 14:07


















What they mean by "$S(t)$ is the subspace where the homogeneous solutions proceed" is that for the homogeneous equation, $x(t)$ has to lie in $S(t)$. This is because otherwise it would not be possible to satisfy the homogeneous equation for any value of $x'(t)$.
– Rahul
Nov 16 at 13:26






What they mean by "$S(t)$ is the subspace where the homogeneous solutions proceed" is that for the homogeneous equation, $x(t)$ has to lie in $S(t)$. This is because otherwise it would not be possible to satisfy the homogeneous equation for any value of $x'(t)$.
– Rahul
Nov 16 at 13:26














@Rahul Sorry, i don't know,That "$S(t)$ is the subspace of the homogeneous equation solutions" is that $forall z in S(t)$ is the solution of $A(t)x'(t)+B(t)x(t)=0$?? And $A(t)z'+B(t)z=0, B(t)z=A(t)y$
– Hewie Ding
Nov 16 at 14:07






@Rahul Sorry, i don't know,That "$S(t)$ is the subspace of the homogeneous equation solutions" is that $forall z in S(t)$ is the solution of $A(t)x'(t)+B(t)x(t)=0$?? And $A(t)z'+B(t)z=0, B(t)z=A(t)y$
– Hewie Ding
Nov 16 at 14:07

















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