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?
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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
asked Nov 23 at 6:16
learning
255
add a comment |
up vote
0
down vote
favorite
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
asked Nov 23 at 6:16
learning
255
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
add a comment |
up vote
0
down vote
favorite
up vote
0
down vote
favorite
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
asked Nov 23 at 6:16
learning
255
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
linear-algebra
asked Nov 23 at 6:16
learning
255
asked Nov 23 at 6:16
learning
255
asked Nov 23 at 6:16
learning
255
asked Nov 23 at 6:16
learning
255
asked Nov 23 at 6:16
learning
255
255
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
add a comment |
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
add a comment |
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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