ABSTRACT
In engineering mechanics, we have worked with scalars, vectors, matrices, and spaces such as work, forces, stiffness matrices, and the three-dimensional Euclidean space. Then, what is a vector space? In addition, what is required to define a vector space? A vector space is a mathematical concept. Without use of rigorous mathematical definitions, we can state that a vector space is a nonempty set V of elements or vectors a, b, c, …, in which two algebraic operations called vector addition and vector multiplication by scalars are defined. Clearly, if the scalars for vector multiplication are real numbers, a real vector space is obtained; if the scalars are complex numbers, we have a complex vector space. In computational structural mechanics, we usually deal with real vector spaces. In some cases, we need to work with complex vector spaces. In general, a vector space includes sets of vectors, matrices, functions, and operators. The concept of vector space is fundamental in functional analysis, which has direct implications in finite element analysis.
