Is a linear voltage regulator necessary in AC to DC?
The rectifier alone produces very rough DC; it is not smooth at all and contains much of the AC signal called "ripple." 



A filtering capacitor helps to smooth this out some, but the output is still not perfectly invariant. 



The addition of the 7812 in this schematic, regulates that DC down to 12v, and eliminates almost all of the remaining ripple, producing a very nice and clean DC output.

There is nothing stating that you must use a regulator; and in fact, they are sometimes not used. Such as if the power supply were just lighting a lamp or operating a fan. Those "loads" do not require precise power control. But for more sensitive electronics such as digital IC's and analog chips, their supply voltage must remain very steady for correct operation. So the answer is, whether or not to use a regulator depends on the application. :)