How to determine linear transformations from $R^n$ to $R^m$ is one-to-one, if $m≠n$ I know if $T:R^n → R^n$ is linear operator then, following two statements are equivalent
a) $[T]$ is invertible matrix, where $[T]$ is matrix representation of $T$
b) $T$ is one-to-one operator. 
Now, if i consider $T: R^n → R^m$ where $m ≠ n$ then how to verify given transformation is one-to-one? 
Is I need to take the help of Rank nullity theorem? First I need to find $[T]$ i. e. Matrix representation of $T$ then, first I need to find $rank([T])$ and by using rank-nullity theorem,
$rank([T]) + nullity([T]) = n$
→ $nullity([T]) = n - rank([T])$
If $nullity([T]) = 0$ then $T$ is one-to-one transformations form $R^n$ to $R^m$
Is am i correct? If i am correct, i need reason behind this? Please I need help? 
 A: The first part of your post can be generalized by the following theorem.

Theorem: Let $V$ and $W$ be finite dimensional vector spaces with $dim(V)=dim(W)$ and let $T:V\to W$ be linear. Then the following are equivalent.
(1) $T$ is one-one.
(2) $T$ is onto.
(3) $Rank(T)=dim(V)$

Now coming to your question, "What happens when we have two Vector Spaces with unequal dimensions?When can we say a Linear Transformation between two such Vector Spaces is $one-one?$"
Well consider the following result,

Theorem: Let $V$ and $W$ be Vector Spaces and let $T:V\to W$ be linear. Then $T$ is $one-one$ $\iff Ker(T)=\{0\}$

Proof:
Let $T$ be $one-one$.
If $x\in Ker(T)$, then $T(x)=0=T(0)\implies x=0\;\;\;\;\;\;\;\text{[Since T is one-one]}$.
Thus $Ker(T)=\{0\}$
Conversely, let $Ker(T)=\{0\}$.
Consider, $T(x)=T(y)\implies T(x)-T(y)=0\implies T(x-y)=0 \implies x-y=0\implies x=y.$
Thus $T$ is $one-one.$
For computational purpose, we usually find the matrix of Linear Transformation and then perform the row reductions to obtain the nullity.
I would like to end the answer with the following question for you, "What guarantees the existence of such a matrix for each Linear Transformation? Or in other words can there be a Linear Transformation with no matrix representation?"
