In terms of transformations, which characteristics do projections lack?

Projections in linear algebra are specific types of linear transformations that map vectors onto a subspace. One of the defining characteristics of projections is that they are not invertible. This means that if you take a vector and project it onto a subspace, there isn't a unique way to recover the original vector from its projection. Specifically, multiple original vectors can yield the same projection, since different points in the space outside the subspace may project down to the same point in the subspace.

Invertibility requires that each element of the image (the outcome of the transformation) corresponds to a unique element in the domain. For projections, this is violated because there is an inherent loss of information as dimensions are effectively reduced. In other words, once you project a vector onto a lower-dimensional space, you lose the components of the vector that were orthogonal to that space, making it impossible to retrieve the original vector.

The other characteristics—completeness, consistency, and orthogonality—are not inherently lacking in projections in the same way. Completeness can refer to whether the subspace spans the entire space, which does not apply directly to the concept of projection itself. Consistency can be interpreted in a couple of ways but generally does relate to the behavior