For matrix transformations, what does 'invertibility' suggest about the transformation?

Invertibility in the context of matrix transformations indicates that for every output in the transformation, there is a unique input that corresponds to it. This means that the transformation can be "reversed"; if you have the transformation ( T ) represented by a matrix ( A ), being invertible implies that there is another matrix ( A^{-1} ) such that when you apply it to the output of ( T ) (i.e., ( T(x) = Ax )), you can retrieve the original input ( x ).

In practical terms, this unique mapping property signifies that the transformation is one-to-one (injective) and onto (surjective) in its action over the space it transforms, preserving the uniqueness of each mapped point. Thus, no two different inputs yield the same output, which is central to the concept of invertibility.

The other options do not correctly reflect the implications of invertibility. For instance, linearity is preserved regardless of invertibility; scaling vectors refers to a different type of transformation, such as dilation or contraction, and changing dimensionality pertains to transformations that may result in a loss or increase of rank, which is not necessarily indicative of whether a transformation is invertible.