What does it mean if a transformation matrix is singular?

A transformation matrix is considered singular if it does not have an inverse. This condition arises when the matrix has a determinant of zero, indicating that the linear transformation it represents is not one-to-one. This means that at least one set of distinct input vectors may map to the same output vector, leading to the conclusion that it cannot transform vectors to a distinct output. In practical terms, this results in a loss of information, where certain dimensions in the input space are collapsed, and multiple inputs are effectively treated as the same output.

Other concepts mentioned in the options can help clarify why the choice is particularly relevant. A transformation that has unique outputs would suggest the matrix is non-singular since it would imply every input vector translates to a distinct, identifiable output. A singular matrix also cannot represent a linear transformation that is invertible, which directly contradicts the notion of having an inverse. Non-linearity of a transformation matrix does not inherently pertain to the concept of singularity, as singularity specifically relates to properties of linear transformations represented by matrices. Thus, the ability of a singular transformation matrix to collapse output dimensions directly leads to multiple inputs producing the same output, solidifying the correctness of the choice.