Correcting a question so that it becomes true. Announcing the arrival of Valued Associate #679: Cesar Manara Planned maintenance scheduled April 23, 2019 at 23:30 UTC (7:30pm US/Eastern)Prove or disprove: the Hilbert-Schmidt norm is independent of the choice of basis on $mathbbR^n$Prove that there exists r > 0 such that Df(x) is surjective.correcting a mistake in SpivakGiven a matrix $A$, show that bases exist in two vector spaces such that a transformation has $A$ as transformation matrixunder change of coordinates the variety $Z(H_1,..,H_r)$ becomes $Z(x_1,…,x_r)subset mathbbP^n$.$TS=0$ if and only if $T(v)=0$Prove or disprove statement about linear transformationProve that there does not exist a continuous function over reals for which it takes on every value exactly twiceDirect sum, null space and rangeA question about Linear transformations on matricesProving a property of the derivative $ Df(c)$ being surjective linear transformationProve that there exists r > 0 such that Df(x) is surjective.
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Correcting a question so that it becomes true.
Announcing the arrival of Valued Associate #679: Cesar Manara
Planned maintenance scheduled April 23, 2019 at 23:30 UTC (7:30pm US/Eastern)Prove or disprove: the Hilbert-Schmidt norm is independent of the choice of basis on $mathbbR^n$Prove that there exists r > 0 such that Df(x) is surjective.correcting a mistake in SpivakGiven a matrix $A$, show that bases exist in two vector spaces such that a transformation has $A$ as transformation matrixunder change of coordinates the variety $Z(H_1,..,H_r)$ becomes $Z(x_1,…,x_r)subset mathbbP^n$.$TS=0$ if and only if $T(v)=0$Prove or disprove statement about linear transformationProve that there does not exist a continuous function over reals for which it takes on every value exactly twiceDirect sum, null space and rangeA question about Linear transformations on matricesProving a property of the derivative $ Df(c)$ being surjective linear transformationProve that there exists r > 0 such that Df(x) is surjective.
$begingroup$
The question is given here in the following link:
Prove that there exists r > 0 such that Df(x) is surjective.
But I am wondering if I change the statement from "there exist r" to "for all r" will the statement be correct?
Also what does "Df(c) is surjective linear transformation" in the statement of the question mean in the 1-dimensional case?
real-analysis calculus linear-algebra analysis multivariable-calculus
asked Apr 2 at 1:56
hopefullyhopefully
192215
$endgroup$
|
show 18 more comments
$begingroup$
The question is given here in the following link:
Prove that there exists r > 0 such that Df(x) is surjective.
But I am wondering if I change the statement from "there exist r" to "for all r" will the statement be correct?
Also what does "Df(c) is surjective linear transformation" in the statement of the question mean in the 1-dimensional case?
real-analysis calculus linear-algebra analysis multivariable-calculus
asked Apr 2 at 1:56
hopefullyhopefully
192215
$endgroup$
2
$begingroup$
Am I misunderstanding - if you can't prove it true for a single $r$, how can you prove it for all $r$? And for functions $f:mathbb Rtomathbb R$, the linear map $df(x) : mathbb Rtomathbb R$ is surjective iff it is invertible (i.e. not zero)
$endgroup$
– Calvin Khor
Apr 2 at 2:03
1
$begingroup$
It is like the linked counterexample proves that $f(n)$ is not divisible by 3, and now you are asking if $f(n)$ is divisible by 9. Suppose you could prove the "for all $r$" version is true. In particular then, it is true for $r=1$ and therefore there exists an $r$ where it is true, in contradiction with the linked counterexample
$endgroup$
– Calvin Khor
Apr 2 at 2:30
1
$begingroup$
"but $f'(x)$ takes all real values (including $0$) in any neighbourhood of $0$."
$endgroup$
– Calvin Khor
Apr 2 at 2:48
1
$begingroup$
the answer is indeed a counterexample to the linked question
$endgroup$
– Calvin Khor
Apr 2 at 2:52
1
$begingroup$
It seems you understand, so I dont think you need to ask me that?
$endgroup$
– Calvin Khor
Apr 2 at 2:55
|
show 18 more comments
$begingroup$
The question is given here in the following link:
Prove that there exists r > 0 such that Df(x) is surjective.
But I am wondering if I change the statement from "there exist r" to "for all r" will the statement be correct?
Also what does "Df(c) is surjective linear transformation" in the statement of the question mean in the 1-dimensional case?
real-analysis calculus linear-algebra analysis multivariable-calculus
asked Apr 2 at 1:56
hopefullyhopefully
192215
$endgroup$
The question is given here in the following link:
Prove that there exists r > 0 such that Df(x) is surjective.
But I am wondering if I change the statement from "there exist r" to "for all r" will the statement be correct?
Also what does "Df(c) is surjective linear transformation" in the statement of the question mean in the 1-dimensional case?
real-analysis calculus linear-algebra analysis multivariable-calculus
real-analysis calculus linear-algebra analysis multivariable-calculus
asked Apr 2 at 1:56
hopefullyhopefully
192215
asked Apr 2 at 1:56
hopefullyhopefully
192215
asked Apr 2 at 1:56
hopefullyhopefully
192215
asked Apr 2 at 1:56
hopefullyhopefully
192215
asked Apr 2 at 1:56
hopefullyhopefully
192215
192215
2
$begingroup$
Am I misunderstanding - if you can't prove it true for a single $r$, how can you prove it for all $r$? And for functions $f:mathbb Rtomathbb R$, the linear map $df(x) : mathbb Rtomathbb R$ is surjective iff it is invertible (i.e. not zero)
$endgroup$
– Calvin Khor
Apr 2 at 2:03
1
$begingroup$
It is like the linked counterexample proves that $f(n)$ is not divisible by 3, and now you are asking if $f(n)$ is divisible by 9. Suppose you could prove the "for all $r$" version is true. In particular then, it is true for $r=1$ and therefore there exists an $r$ where it is true, in contradiction with the linked counterexample
$endgroup$
– Calvin Khor
Apr 2 at 2:30
1
$begingroup$
"but $f'(x)$ takes all real values (including $0$) in any neighbourhood of $0$."
$endgroup$
– Calvin Khor
Apr 2 at 2:48
1
$begingroup$
the answer is indeed a counterexample to the linked question
$endgroup$
– Calvin Khor
Apr 2 at 2:52
1
$begingroup$
It seems you understand, so I dont think you need to ask me that?
$endgroup$
– Calvin Khor
Apr 2 at 2:55
|
show 18 more comments
2
$begingroup$
Am I misunderstanding - if you can't prove it true for a single $r$, how can you prove it for all $r$? And for functions $f:mathbb Rtomathbb R$, the linear map $df(x) : mathbb Rtomathbb R$ is surjective iff it is invertible (i.e. not zero)
$endgroup$
– Calvin Khor
Apr 2 at 2:03
1
$begingroup$
It is like the linked counterexample proves that $f(n)$ is not divisible by 3, and now you are asking if $f(n)$ is divisible by 9. Suppose you could prove the "for all $r$" version is true. In particular then, it is true for $r=1$ and therefore there exists an $r$ where it is true, in contradiction with the linked counterexample
$endgroup$
– Calvin Khor
Apr 2 at 2:30
1
$begingroup$
"but $f'(x)$ takes all real values (including $0$) in any neighbourhood of $0$."
$endgroup$
– Calvin Khor
Apr 2 at 2:48
1
$begingroup$
the answer is indeed a counterexample to the linked question
$endgroup$
– Calvin Khor
Apr 2 at 2:52
1
$begingroup$
It seems you understand, so I dont think you need to ask me that?
$endgroup$
– Calvin Khor
Apr 2 at 2:55
2
2
$begingroup$
Am I misunderstanding - if you can't prove it true for a single $r$, how can you prove it for all $r$? And for functions $f:mathbb Rtomathbb R$, the linear map $df(x) : mathbb Rtomathbb R$ is surjective iff it is invertible (i.e. not zero)
$endgroup$
– Calvin Khor
Apr 2 at 2:03
$begingroup$
Am I misunderstanding - if you can't prove it true for a single $r$, how can you prove it for all $r$? And for functions $f:mathbb Rtomathbb R$, the linear map $df(x) : mathbb Rtomathbb R$ is surjective iff it is invertible (i.e. not zero)
$endgroup$
– Calvin Khor
Apr 2 at 2:03
1
1
$begingroup$
It is like the linked counterexample proves that $f(n)$ is not divisible by 3, and now you are asking if $f(n)$ is divisible by 9. Suppose you could prove the "for all $r$" version is true. In particular then, it is true for $r=1$ and therefore there exists an $r$ where it is true, in contradiction with the linked counterexample
$endgroup$
– Calvin Khor
Apr 2 at 2:30
$begingroup$
It is like the linked counterexample proves that $f(n)$ is not divisible by 3, and now you are asking if $f(n)$ is divisible by 9. Suppose you could prove the "for all $r$" version is true. In particular then, it is true for $r=1$ and therefore there exists an $r$ where it is true, in contradiction with the linked counterexample
$endgroup$
– Calvin Khor
Apr 2 at 2:30
1
1
$begingroup$
"but $f'(x)$ takes all real values (including $0$) in any neighbourhood of $0$."
$endgroup$
– Calvin Khor
Apr 2 at 2:48
$begingroup$
"but $f'(x)$ takes all real values (including $0$) in any neighbourhood of $0$."
$endgroup$
– Calvin Khor
Apr 2 at 2:48
1
1
$begingroup$
the answer is indeed a counterexample to the linked question
$endgroup$
– Calvin Khor
Apr 2 at 2:52
$begingroup$
the answer is indeed a counterexample to the linked question
$endgroup$
– Calvin Khor
Apr 2 at 2:52
1
1
$begingroup$
It seems you understand, so I dont think you need to ask me that?
$endgroup$
– Calvin Khor
Apr 2 at 2:55
$begingroup$
It seems you understand, so I dont think you need to ask me that?
$endgroup$
– Calvin Khor
Apr 2 at 2:55
|
show 18 more comments
0
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$begingroup$
Am I misunderstanding - if you can't prove it true for a single $r$, how can you prove it for all $r$? And for functions $f:mathbb Rtomathbb R$, the linear map $df(x) : mathbb Rtomathbb R$ is surjective iff it is invertible (i.e. not zero)
$endgroup$
– Calvin Khor
Apr 2 at 2:03
1
$begingroup$
It is like the linked counterexample proves that $f(n)$ is not divisible by 3, and now you are asking if $f(n)$ is divisible by 9. Suppose you could prove the "for all $r$" version is true. In particular then, it is true for $r=1$ and therefore there exists an $r$ where it is true, in contradiction with the linked counterexample
$endgroup$
– Calvin Khor
Apr 2 at 2:30
1
$begingroup$
"but $f'(x)$ takes all real values (including $0$) in any neighbourhood of $0$."
$endgroup$
– Calvin Khor
Apr 2 at 2:48
1
$begingroup$
the answer is indeed a counterexample to the linked question
$endgroup$
– Calvin Khor
Apr 2 at 2:52
1
$begingroup$
It seems you understand, so I dont think you need to ask me that?
$endgroup$
– Calvin Khor
Apr 2 at 2:55