How to determine linear transformations from $R^n$ to $R^m$ is one-to-one, if $m≠n$

Question

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?

Answer 1

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?"