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So every process has its own Page Table, the page table references the frame where the page is in physical memory and also has a valid-invalid bit that tells whether it is in physical memory or in secondary memory.
But how does the S.O. knows exactly where the page is saved when the valid-invalid bit it's off? Is there another table I'm missing? Or it just does a linear search through the whole SWAP space?
The first thing is that when the page-table entry is invalid you are allowed to use all the other bits in the page-table entry for whatever you want. So, for example, you could use it to store an offset into the swap space, or a disk block number.
But that's actually not the whole story. Each process on most operating systems has some kind of segment map that describes for each valid range of logical addresses what's in that range. Ranges that contain read-only data (such as the code segment, usually) don't need to be swapped. Instead they can just be reread from the executable that contains them if they are needed again.
