Suppose that $\ltdefn{T}{U}{V}$ is an injective linear transformation. Then $\dimension{U}\leq\dimension{V}$.
Notice that the previous example made no use of the actual formula defining the function. Merely a comparison of the dimensions of the domain and codomain are enough to conclude that the linear transformation is not injective. Archetype M and Archetype N are two more examples of linear transformations that have “big” domains and “small” codomains, resulting in “collisions” of outputs and thus are non-injective linear transformations.