Introduction to Proofs and Proof Strategies

Linear Algebra is a branch of mathematics that deals with linear equation, systems of linear equations, and their representation as functions between algebraic structures called vector spaces. Linear Algebra is an essential tool in many disciplines such as engineering, statistics, and computer science and is also central within mathematics, in areas such as analysis and geometry. Much like the definition of a field, a vector space is defined through a list of axioms, motivated by concrete observations in familiar spaces, such as the standard two-dimensional plane and the three-dimensional space. We begin this chapter by taking a closer look at real n-dimensional spaces and vectors and then move on to discussing abstract vector spaces and linear maps. Our experience with sets and functions, developed in previous chapters, as well as certain proof techniques, will prove to be useful in our discussion.