In this chapter, we will first study the real vector space of \(2\times 2\) Hermitian matrices. This is a vector space although its elements are matrices. We will show that any \(2\times 2\) Hermitian matrix can be represented as a linear combination of the Pauli matrices and the identity matrix. We will then introduce the inner product of matrices, which is just a natural extension of the concept we learned in a regular vector space. Then we will discuss the concept of density matrix, which is very useful for describing mixed states due to the lack of information or the system being entangled with the external environment. Finally, we will show how to find the expectation values of an operator using a density matrix. Particularly, when the operator is a linear combination of the Pauli matrices, its expectation value is just the projection of the state (including mixed state) on the corresponding normalized Bloch vector on the Bloch sphere.