A basis for a vector space has two properties: the vectors in the basis span the space and they are independent. It is easy to see how to write any "vector" as a "linear combination" of thex_nand it should be just as easy to see that they are independent.

(At first I missed the "in F2" which is only in your title, not in the body of the question, and wondered about scalar multiplication. Of course, the only members of F2 are 0 and 1. 0 times any vector is 0 and 1 times any vector is the vector itself.)