If matrix $A in M_{m,n}$ and $n geq m$, matrix $X in M_{n,k}$, then when $A X = 0$ (nullspace) is true?












0














Please excuse me asking for the fundamental rank-nullity related problem.




If matrix $A in M_{m,n}$ and $n geq m$, matrix $X in M_{n,k}$, then when $A X = 0$ (nullspace) is true?




Can this be proved rigorously?





My thinking:



At least (necessary? and sufficient condition?) $n$ must be greater than $m$, that is $n > m$ such that $AX=0$. Am I thinking correct? or completely off-the track? But I don't know how to prove it rigorously Thank you so much.










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  • 1




    I think you're off-track. Necessary & sufficient is that each column of $X$ be in the nullspace of $A$.
    – Gerry Myerson
    Nov 23 at 6:35










  • Thanks for the clarification, Gerry! However, what are the requirements on the matrix $A$ and also $X$ such that the column of $X$ lie in the $mathcal{N}(A)$? I hope my question makes sense to you.
    – learning
    Nov 23 at 9:56






  • 1




    I don't know of any better way of requiring the columns of $X$ to lie in the nullspace of $A$, than to require that the columns of $X$ lie in the nullspace of $A$. I suppose you could say every column of $X$ has to be orthogonal to every row of $A$ – is that any better? What kind of answer are you hoping for?
    – Gerry Myerson
    Nov 23 at 10:16










  • I get it now. Thank you very much, Gerry.
    – learning
    Nov 23 at 10:21
















0














Please excuse me asking for the fundamental rank-nullity related problem.




If matrix $A in M_{m,n}$ and $n geq m$, matrix $X in M_{n,k}$, then when $A X = 0$ (nullspace) is true?




Can this be proved rigorously?





My thinking:



At least (necessary? and sufficient condition?) $n$ must be greater than $m$, that is $n > m$ such that $AX=0$. Am I thinking correct? or completely off-the track? But I don't know how to prove it rigorously Thank you so much.










share|cite|improve this question


















  • 1




    I think you're off-track. Necessary & sufficient is that each column of $X$ be in the nullspace of $A$.
    – Gerry Myerson
    Nov 23 at 6:35










  • Thanks for the clarification, Gerry! However, what are the requirements on the matrix $A$ and also $X$ such that the column of $X$ lie in the $mathcal{N}(A)$? I hope my question makes sense to you.
    – learning
    Nov 23 at 9:56






  • 1




    I don't know of any better way of requiring the columns of $X$ to lie in the nullspace of $A$, than to require that the columns of $X$ lie in the nullspace of $A$. I suppose you could say every column of $X$ has to be orthogonal to every row of $A$ – is that any better? What kind of answer are you hoping for?
    – Gerry Myerson
    Nov 23 at 10:16










  • I get it now. Thank you very much, Gerry.
    – learning
    Nov 23 at 10:21














0












0








0







Please excuse me asking for the fundamental rank-nullity related problem.




If matrix $A in M_{m,n}$ and $n geq m$, matrix $X in M_{n,k}$, then when $A X = 0$ (nullspace) is true?




Can this be proved rigorously?





My thinking:



At least (necessary? and sufficient condition?) $n$ must be greater than $m$, that is $n > m$ such that $AX=0$. Am I thinking correct? or completely off-the track? But I don't know how to prove it rigorously Thank you so much.










share|cite|improve this question













Please excuse me asking for the fundamental rank-nullity related problem.




If matrix $A in M_{m,n}$ and $n geq m$, matrix $X in M_{n,k}$, then when $A X = 0$ (nullspace) is true?




Can this be proved rigorously?





My thinking:



At least (necessary? and sufficient condition?) $n$ must be greater than $m$, that is $n > m$ such that $AX=0$. Am I thinking correct? or completely off-the track? But I don't know how to prove it rigorously Thank you so much.







linear-algebra






share|cite|improve this question













share|cite|improve this question











share|cite|improve this question




share|cite|improve this question










asked Nov 23 at 6:16









learning

275




275








  • 1




    I think you're off-track. Necessary & sufficient is that each column of $X$ be in the nullspace of $A$.
    – Gerry Myerson
    Nov 23 at 6:35










  • Thanks for the clarification, Gerry! However, what are the requirements on the matrix $A$ and also $X$ such that the column of $X$ lie in the $mathcal{N}(A)$? I hope my question makes sense to you.
    – learning
    Nov 23 at 9:56






  • 1




    I don't know of any better way of requiring the columns of $X$ to lie in the nullspace of $A$, than to require that the columns of $X$ lie in the nullspace of $A$. I suppose you could say every column of $X$ has to be orthogonal to every row of $A$ – is that any better? What kind of answer are you hoping for?
    – Gerry Myerson
    Nov 23 at 10:16










  • I get it now. Thank you very much, Gerry.
    – learning
    Nov 23 at 10:21














  • 1




    I think you're off-track. Necessary & sufficient is that each column of $X$ be in the nullspace of $A$.
    – Gerry Myerson
    Nov 23 at 6:35










  • Thanks for the clarification, Gerry! However, what are the requirements on the matrix $A$ and also $X$ such that the column of $X$ lie in the $mathcal{N}(A)$? I hope my question makes sense to you.
    – learning
    Nov 23 at 9:56






  • 1




    I don't know of any better way of requiring the columns of $X$ to lie in the nullspace of $A$, than to require that the columns of $X$ lie in the nullspace of $A$. I suppose you could say every column of $X$ has to be orthogonal to every row of $A$ – is that any better? What kind of answer are you hoping for?
    – Gerry Myerson
    Nov 23 at 10:16










  • I get it now. Thank you very much, Gerry.
    – learning
    Nov 23 at 10:21








1




1




I think you're off-track. Necessary & sufficient is that each column of $X$ be in the nullspace of $A$.
– Gerry Myerson
Nov 23 at 6:35




I think you're off-track. Necessary & sufficient is that each column of $X$ be in the nullspace of $A$.
– Gerry Myerson
Nov 23 at 6:35












Thanks for the clarification, Gerry! However, what are the requirements on the matrix $A$ and also $X$ such that the column of $X$ lie in the $mathcal{N}(A)$? I hope my question makes sense to you.
– learning
Nov 23 at 9:56




Thanks for the clarification, Gerry! However, what are the requirements on the matrix $A$ and also $X$ such that the column of $X$ lie in the $mathcal{N}(A)$? I hope my question makes sense to you.
– learning
Nov 23 at 9:56




1




1




I don't know of any better way of requiring the columns of $X$ to lie in the nullspace of $A$, than to require that the columns of $X$ lie in the nullspace of $A$. I suppose you could say every column of $X$ has to be orthogonal to every row of $A$ – is that any better? What kind of answer are you hoping for?
– Gerry Myerson
Nov 23 at 10:16




I don't know of any better way of requiring the columns of $X$ to lie in the nullspace of $A$, than to require that the columns of $X$ lie in the nullspace of $A$. I suppose you could say every column of $X$ has to be orthogonal to every row of $A$ – is that any better? What kind of answer are you hoping for?
– Gerry Myerson
Nov 23 at 10:16












I get it now. Thank you very much, Gerry.
– learning
Nov 23 at 10:21




I get it now. Thank you very much, Gerry.
– learning
Nov 23 at 10:21















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