If $T:\mathbb{R}^n \rightarrow \mathbb{R}^m$ is a matrix transformation, does $T$ depend on the dimensions of $\mathbb{R}$?
i.e., is $T$ one-one if $m>n$, $m=n$, or $n>m$?
Also, say if $T$ is one-one, does this mean it is a matrix transformation and hence a linear transformation?
Any matrix transformation $T:\mathbb{R}^n\to\mathbb{R}^m$ is a linear transformation (and vice versa once you've specified bases). If $n>m$, then $T$ cannot be injective because there cannot be $n$ linearly independent vectors in $\mathbb{R}^m$. Finally, if $n=m$ or $n<m$, one can say nothing about injectivity without more information. For instance, for $n=m$ you have the zero matrix transformation (not injective) and the identity matrix transformation (injective).