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If $T:{\mathbb{R}}^{n}\to {\mathbb{R}}^{m}$ is a matrix transformation, does $Tn,m=n, orn>m?$
Also, say if $T$ is one-one, does this mean it is a matrix transformation and hence a linear transformation?
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Monserrat Cole
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, 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.