1. ## Linear operator

Please can you help me to track the question below.
Q. Let T:X-->Y be a linear operator and dim X= dim Y = n <\infty . show that range R(T)=Y if and only if T^-1 exist.

2. Originally Posted by kinkong
Please can you help me to track the question below.
Q. Let T:X-->Y be a linear operator and dim X= dim Y = n <\infty . show that range R(T)=Y if and only if T^-1 exist.

Hint

$\displaystyle \textrm{r}(T)=Y\Leftrightarrow \dim\textrm{r}(T)=n\Leftrightarrow \dim\ker(T)=0$

3. gee, does everything in linear algebra reduce to the rank-nullity theorem?

4. Originally Posted by Deveno
gee, does everything in linear algebra reduce to the rank-nullity theorem?

Everything related with that theorem.

5. your hint was useful but i couldnt figure where to go from there. please can you help me out...thanks

6. Originally Posted by kinkong
your hint was useful but i couldnt figure where to go from there. please can you help me out...thanks

$\displaystyle T$ is injective iff $\displaystyle \ker T=\{0\}$ and $\displaystyle T$ is surjective iff $\displaystyle r(T)=Y$ . Besides, there exists $\displaystyle T^{-1}$ iff $\displaystyle T$ is bijective.