ABSTRACT
In this chapter, we will review the theory about vectors and linear spaces. We will provide a framework to characterize linear spaces which are not only of dimension two or three, but of possibly much larger dimension. These spaces tend to be somewhat abstract, but they are a powerful concept in dealing with many real-life problems,
We first define a general linear space L. We denote its elements-the vectors-by boldface letters such as u, v, etc. One basic operation must be defined. It is the linear combination of two vectors su + tv with scalars s and t. With this operation in place, the defining property for a linear space is that any linear combination of vectors results in a vector in the same space. More precisely, if v1,v2, . . . ,vn are in L then any linear combination v of the form
v = s1v1 + s2v2 + . . .+ snvn (15.1)
is also in L. Note that all si may be zero, asserting that every linear space has a zero vector in it.
