What is the difference between linear subspace and affine subspace in PCA?

Linear subspaces pass through origin; affine subspaces are shifted versions of linear subspaces.

PCA effectively finds a linear subspace after centering the data. If data is not centered, PCA implicitly captures an affine subspace, which includes a translation component. This distinction is important because variance decomposition assumes origin-centered geometry; otherwise, the mean shift contaminates principal directions.