# Thread: Prove or disprove question

1. ## Prove or disprove question

Prove or dispove

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

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

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

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!

2. Originally Posted by jayshizwiz
Prove or dispove

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

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

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

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?

3. if $\displaystyle T(u) \cdot v = 0$

that must mean

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

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

Is there any error in my reasoning?

Thanks!

4. Does anybody have an idea? Is the question clear?