Prove or disprove question

**jayshizwiz** · November 7th 2010, 07:47 AM

Prove or dispove

$T:R^n \rightarrow R^n$ is a linear transformation

if for every $u \in R^n$ and for every $v \in KerT$ ,
$T(u) \cdot v=0$

then $KerT = (ImT)^{\perp}$

Attempt:

even though these questions are always false, sadly I chose True and gave an explanation why.

If this is false can someone please explain why and give an example.
(even if it's true can you explain why)

Thanks!

**Drexel28** · November 7th 2010, 08:05 AM

Originally Posted by jayshizwiz

Prove or dispove

$T:R^n \rightarrow R^n$ is a linear transformation

if for every $u \in R^n$ and for every $v \in KerT$ ,
$T(u) \cdot v=0$

then $KerT = (ImT)^{\perp}$

Attempt:

even though these questions are always false, sadly I chose True and gave an explanation why.

If this is false can someone please explain why and give an example.
(even if it's true can you explain why)

Thanks!

Would it maybe not be more helpful to you, if you wrote out the reason why you believe it to be true?

**jayshizwiz** · November 7th 2010, 08:19 AM

if $T(u) \cdot v = 0$

that must mean

a) $KerT=\bar{0}$ and $ImT = R^n$
or
b) $KerT=R^n$ and $ImT=\bar{0}$

and in both those cases $kerT= (ImT)^\perp$

Is there any error in my reasoning?

Thanks!

**jayshizwiz** · November 7th 2010, 09:19 PM

Does anybody have an idea? Is the question clear?

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