Cs 798: Theory of Quantum Information Linear Algebra and Analysis 1 Linear Algebra

The purpose of this first set of lecture notes is to summarize background material on linear algebra and analysis that is used throughout the course. Proofs of most of the stated facts may be found in the references listed at the end of the notes. It is my intention that this first set of notes will mostly serve as a useful reference later in the course. The theory of quantum information is based on linear algebra, and in particular the study of finite dimensional vector spaces over the real or complex numbers. We will mostly focus on complex vector spaces, but sometimes it will be necessary to work with real vector spaces. Vector spaces over the real or complex numbers will be denoted by scripted capital letters in this course, generally near the end of the alphabet such as X , Y, and Z. Elements of these spaces will be denoted by lowercase Roman letters such as u, v, w and x. Subsets of vector spaces, which may not themselves be vector spaces, will also be denoted by scripted letters, usually near the beginning of the alphabet such as A, B, and C. The field of real numbers is denoted R and the field of complex numbers is denoted C. The first two properties imply conjugate linearity in the first argument: A norm on a complex vector space X over C is a function ν : X → R that satisfies the following properties. 1. Positive definiteness: ν(u) > 0 for all u ∈ X \{0} and ν(0) = 0. 2. Positive scalability: ν(αu) = |α| ν(u) for all u ∈ X and α ∈ C. 3. The Triangle Inequality: ν(u + v) ≤ ν(u) + ν(v) for all u, v ∈ X .