  Linear regression is a classic method of data analysis. In recent years,
sketching -- a method of dimension reduction using random sampling, random
projections, or both -- has gained popularity as an effective computational
approximation when the number of observations greatly exceeds the number of
variables. In this paper, we address the following question: How does sketching
affect the statistical properties of the solution and key quantities derived
from it?
  To answer this question, we present a projector-based approach to sketched
linear regression that is exact and that requires minimal assumptions on the
sketching matrix. Therefore, downstream analyses hold exactly and generally for
all sketching schemes. Additionally, a projector-based approach enables
derivation of key quantities from classic linear regression that account for
the combined model- and algorithm-induced uncertainties. We demonstrate the
usefulness of a projector-based approach in quantifying and enabling insight on
excess uncertainties and bias-variance decompositions for sketched linear
regression. Finally, we demonstrate how the insights from our projector-based
analyses can be used to produce practical sketching diagnostics to aid the
design of judicious sketching schemes.
