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Explaining
F_2 = the 1-dimensional Vector Space over the scalar field with 2 elements.
via actions.
Vectors=Actions
States=What is reached by applying an action to a state
Eventually Vectors and States can be identified. Simply call the state where you start 0 and v the state that is reachable from 0 by applying the action v to state 0.
I.e. 0+v=v