Professor Susskind elaborates on the abstract mathematics of vector spaces by introducing the concepts of basis vectors, linear combinations of vector states, and matrix algebra as it applies to vector spaces. He then introduces linear operators and bra-ket notation, and presents Hermitian operators as a special class of operators that represent observables. Eigenvectors of Hermitian operators represent orthogonal vector states, and their eigenvalues are the values of the observable. Professor Susskind then applies these concepts to the single spin system that we studied in the last lecture, and introduces the Pauli matrices as the Hermitian operators representing the three spin axis directions.