Does PCA preserve linear separability for every linearly separable set? After using PCA to reduce dimensionality, does PCA preserve linear separability for any linearly separable set? Will the data still be linearly separable after the transformation? I am thinking that it does preserve linear separability because the PCA just reduces the dimension and not the relationship between the points in terms of separability. Am I on the right line of thinking?

No, it may be that the discriminative information is in the direction of a principal component that explains a relatively small amount of the total variance, and hence gets discarded.
Consider a two-dimensional dataset where the two classes lie in long parallel cigar-shaped elongated Gaussian clusters with a small gap between them.  Most of the variance lies along the long axis of the clusters, so the first PC will be in that direction and the second will be orthogonal to it.  The data are indistinguishable from the first component, so if you discard the second component, the data will no longer be separable.
