In this and the next few chapters, we will review linear algebra basics. In this chapter, we will review the definitions of vector space, inner product space, and Hilbert space. Although it is not necessary to understand them thoroughly if you feel comfortable with using the equations and mathematical tools given, it is still very instructive if you want to have a basic idea of the foundation of quantum computing and quantum physics. We will also practice \(bra\text{-}ket\) notation and their representations in row and column vectors. We will study how to represent a vector in different bases. Then, we will discuss the effect of measurement in quantum mechanics and the importance of orthonormal basis and normalized vectors. Finally, we will review how to combine two spaces (of two physical systems) into a larger one using the tensor product.