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. PCA targets different target than Linear discriminant analysis (LDA). Both of them actually provide linear transformation, but (rough an quick explanation follows):

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*PCA will set the direction of the first axis of the output space in the direction of the maximum variance of the input data (it does not know the concept of classes per se, all samples are treated equally)

*LDA will set the direction of the first axis of the output space in the direction which has (roughly) the best ratio of inter-class variance vs intra-class variance. For that, it needs a concept of classes.

