What is the complement of the null space??
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Given an $A: R^n rightarrow R^m$ that maps from $x$ to $b$, the null space is the set of $x$s that map to $b=0$.
What is the set of $x$s that map to $bneq0$? Is it $R^n - {nullspace}$. Or doesn't that form a space? And if so, what is the correct answer?
linear-algebra
asked Dec 15 '18 at 14:14
blue_noteblue_note
51448
$endgroup$
add a comment |
$begingroup$
Given an $A: R^n rightarrow R^m$ that maps from $x$ to $b$, the null space is the set of $x$s that map to $b=0$.
What is the set of $x$s that map to $bneq0$? Is it $R^n - {nullspace}$. Or doesn't that form a space? And if so, what is the correct answer?
linear-algebra
asked Dec 15 '18 at 14:14
blue_noteblue_note
51448
$endgroup$
1
$begingroup$
It isn't indeed a subspace. The correct answer depends very much on $A$, there's no general answer, as far as I know.
$endgroup$
– Bernard
Dec 15 '18 at 14:19
$begingroup$
@Bernard: thanks. Is the row space somehow related? I read it's perpendicular to the nullspace, but does that imply something?
$endgroup$
– blue_note
Dec 15 '18 at 14:25
1
$begingroup$
The row space is the column space of the matrix of the dual linear mapping. Maybe that's what they mean in what you've read.
$endgroup$
– Bernard
Dec 15 '18 at 14:28
add a comment |
$begingroup$
Given an $A: R^n rightarrow R^m$ that maps from $x$ to $b$, the null space is the set of $x$s that map to $b=0$.
What is the set of $x$s that map to $bneq0$? Is it $R^n - {nullspace}$. Or doesn't that form a space? And if so, what is the correct answer?
linear-algebra
asked Dec 15 '18 at 14:14
blue_noteblue_note
51448
$endgroup$
Given an $A: R^n rightarrow R^m$ that maps from $x$ to $b$, the null space is the set of $x$s that map to $b=0$.
What is the set of $x$s that map to $bneq0$? Is it $R^n - {nullspace}$. Or doesn't that form a space? And if so, what is the correct answer?
linear-algebra
linear-algebra
asked Dec 15 '18 at 14:14
blue_noteblue_note
51448
asked Dec 15 '18 at 14:14
blue_noteblue_note
51448
asked Dec 15 '18 at 14:14
blue_noteblue_note
51448
asked Dec 15 '18 at 14:14
blue_noteblue_note
51448
asked Dec 15 '18 at 14:14
blue_noteblue_note
51448
51448
1
$begingroup$
It isn't indeed a subspace. The correct answer depends very much on $A$, there's no general answer, as far as I know.
$endgroup$
– Bernard
Dec 15 '18 at 14:19
$begingroup$
@Bernard: thanks. Is the row space somehow related? I read it's perpendicular to the nullspace, but does that imply something?
$endgroup$
– blue_note
Dec 15 '18 at 14:25
1
$begingroup$
The row space is the column space of the matrix of the dual linear mapping. Maybe that's what they mean in what you've read.
$endgroup$
– Bernard
Dec 15 '18 at 14:28
add a comment |
1
$begingroup$
It isn't indeed a subspace. The correct answer depends very much on $A$, there's no general answer, as far as I know.
$endgroup$
– Bernard
Dec 15 '18 at 14:19
$begingroup$
@Bernard: thanks. Is the row space somehow related? I read it's perpendicular to the nullspace, but does that imply something?
$endgroup$
– blue_note
Dec 15 '18 at 14:25
1
$begingroup$
The row space is the column space of the matrix of the dual linear mapping. Maybe that's what they mean in what you've read.
$endgroup$
– Bernard
Dec 15 '18 at 14:28
1
1
$begingroup$
It isn't indeed a subspace. The correct answer depends very much on $A$, there's no general answer, as far as I know.
$endgroup$
– Bernard
Dec 15 '18 at 14:19
$begingroup$
It isn't indeed a subspace. The correct answer depends very much on $A$, there's no general answer, as far as I know.
$endgroup$
– Bernard
Dec 15 '18 at 14:19
$begingroup$
@Bernard: thanks. Is the row space somehow related? I read it's perpendicular to the nullspace, but does that imply something?
$endgroup$
– blue_note
Dec 15 '18 at 14:25
$begingroup$
@Bernard: thanks. Is the row space somehow related? I read it's perpendicular to the nullspace, but does that imply something?
$endgroup$
– blue_note
Dec 15 '18 at 14:25
1
1
$begingroup$
The row space is the column space of the matrix of the dual linear mapping. Maybe that's what they mean in what you've read.
$endgroup$
– Bernard
Dec 15 '18 at 14:28
$begingroup$
The row space is the column space of the matrix of the dual linear mapping. Maybe that's what they mean in what you've read.
$endgroup$
– Bernard
Dec 15 '18 at 14:28
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
The row space is the orthogonal complement of the null space, $(operatorname{row}A)^perp=operatorname{ker}A$, which follows pretty easily from the definition. This is because you dot the rows of $A$ with $x$ to get $b$. So if $b=0$, all the dot products are zero.
The solution set of $Ax=b,,bnot=0$, on the other hand, if nonempty, is the set ${x_0+ymid yinoperatorname{ker}A}$, where $x_0$ is any particular solution.
answered Dec 15 '18 at 16:29
Chris CusterChris Custer
13.1k3827
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add a comment |
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$begingroup$
The row space is the orthogonal complement of the null space, $(operatorname{row}A)^perp=operatorname{ker}A$, which follows pretty easily from the definition. This is because you dot the rows of $A$ with $x$ to get $b$. So if $b=0$, all the dot products are zero.
The solution set of $Ax=b,,bnot=0$, on the other hand, if nonempty, is the set ${x_0+ymid yinoperatorname{ker}A}$, where $x_0$ is any particular solution.
answered Dec 15 '18 at 16:29
Chris CusterChris Custer
13.1k3827
$endgroup$
add a comment |
$begingroup$
The row space is the orthogonal complement of the null space, $(operatorname{row}A)^perp=operatorname{ker}A$, which follows pretty easily from the definition. This is because you dot the rows of $A$ with $x$ to get $b$. So if $b=0$, all the dot products are zero.
The solution set of $Ax=b,,bnot=0$, on the other hand, if nonempty, is the set ${x_0+ymid yinoperatorname{ker}A}$, where $x_0$ is any particular solution.
answered Dec 15 '18 at 16:29
Chris CusterChris Custer
13.1k3827
$endgroup$
add a comment |
$begingroup$
The row space is the orthogonal complement of the null space, $(operatorname{row}A)^perp=operatorname{ker}A$, which follows pretty easily from the definition. This is because you dot the rows of $A$ with $x$ to get $b$. So if $b=0$, all the dot products are zero.
The solution set of $Ax=b,,bnot=0$, on the other hand, if nonempty, is the set ${x_0+ymid yinoperatorname{ker}A}$, where $x_0$ is any particular solution.
answered Dec 15 '18 at 16:29
Chris CusterChris Custer
13.1k3827
$endgroup$
The row space is the orthogonal complement of the null space, $(operatorname{row}A)^perp=operatorname{ker}A$, which follows pretty easily from the definition. This is because you dot the rows of $A$ with $x$ to get $b$. So if $b=0$, all the dot products are zero.
The solution set of $Ax=b,,bnot=0$, on the other hand, if nonempty, is the set ${x_0+ymid yinoperatorname{ker}A}$, where $x_0$ is any particular solution.
answered Dec 15 '18 at 16:29
Chris CusterChris Custer
13.1k3827
edited Dec 15 '18 at 16:51
answered Dec 15 '18 at 16:29
Chris CusterChris Custer
13.1k3827
answered Dec 15 '18 at 16:29
Chris CusterChris Custer
13.1k3827
answered Dec 15 '18 at 16:29
Chris CusterChris Custer
13.1k3827
13.1k3827
add a comment |
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$begingroup$
It isn't indeed a subspace. The correct answer depends very much on $A$, there's no general answer, as far as I know.
$endgroup$
– Bernard
Dec 15 '18 at 14:19
$begingroup$
@Bernard: thanks. Is the row space somehow related? I read it's perpendicular to the nullspace, but does that imply something?
$endgroup$
– blue_note
Dec 15 '18 at 14:25
1
$begingroup$
The row space is the column space of the matrix of the dual linear mapping. Maybe that's what they mean in what you've read.
$endgroup$
– Bernard
Dec 15 '18 at 14:28