The Rajya Sabha (meaning the "Council of States") is the upper house of the Parliament of India. Karnataka elects 12 seats and they are indirectly elected by the state legislators of Karnataka. The number of seats, allocated to the party, are determined by the amount of seats, a party possesses during nomination and the party nominates a member to be voted on. Elections within the state legislatures are held using Single transferable vote with proportional representation.
Alphabetical List of all Members from Karnataka since 1952
Source: Indian government (Rajya Sabha) website
References
List of Rajya Sabha members from Karnataka Wikipedia(Text) CC BY-SA