What can be concluded about projections in linear transformations?

In the context of linear transformations, projections play a specific role in mapping vectors onto a subspace. A projection operator is designed in such a way that it takes a vector and projects it onto a predefined subspace. The key characteristic of a projection is that when you apply the projection a second time to the result, you get the same vector back; this is known as idempotence.

For a linear transformation represented by a projection to be invertible, it must be both one-to-one (injective) and onto (surjective). However, projections onto a subspace that is of lower dimension than the original space consistently yield loss of information about the original vector. As a result, many distinct original vectors can project to the same vector in the subspace, hence failing the injective test. Consequently, because they cannot uniquely reverse the effect of the projection, they are never invertible.

In summary, the nature of projection transformations inherently prevents them from being invertible, reinforcing the conclusion that projections are never invertible.