Linear Spaces

Abstract

The two most basic operations on a given set $\mathbb V$ of functions are addition of any two functions in $\mathbb V$ and multiplication of any function in $\mathbb V$ by a constant. The concept of “vector spaces” to be discussed in some details in the first section of this chapter is to specify the so-called closure properties of the set $\mathbb V$, in that these two operations never create any function outside the set $\mathbb V$. More precisely, the set of constants that are used for multiplication to the functions in $\mathbb V$ must be a “scalar field” denoted by $\mathbb F$, and $\mathbb V$ will be called a vector space over the field $\mathbb F$. The only (scalar) fields considered in this book are the set $\mathbb R$ of real numbers as well as its subset $\mathbb Q$ of rational numbers and its superset $\mathbb C$ of complex numbers. Typical examples of vector spaces over $\mathbb F$ include the vector spaces $\mathbb R ^{m,n}$ and $\mathbb C ^{m,n}$ of $m\times n$ matrices of real numbers and of complex numbers, over the scalar fields $\mathbb F = \mathbb R$ and $\mathbb F = \mathbb C$, respectively, where $m$ and $n$ are integers with $m, n \ge 1$. If $m = 1$, then the vector space of matrices becomes the familiar space of row vectors, while for $n = 1$ we have the familiar space of column vectors.