1. Matrices, Algebra, and Analysis
Society for Industrial and Applied Mathematics eBooks · 1998
There are two approaches to linear algebra, each having its virtues. The first is abstract. A vector space is defined axiomatically as a collection of objects, called vectors, with a sum and a scalar-vector product. As the theory develops, matrices emerge, almost incidentally, as scalar representations of linear transformations. The advantage of this approach is generality. The disadvantage is that the hero of our story, the matrix, has to wait in the wings. The second approach is concrete. Vectors and matrices are defined as arrays of scalars — here arrays of real or complex numbers. Operations between vectors and matrices are defined in terms of the scalars that compose them. The advantage of this approach for a treatise on matrix computations is obvious: it puts the objects we are going to manipulate to the fore. Moreover, it is truer to the history of the subject. Most decompositions we use today to solve matrix problems originated as simplifications of quadratic and bilinear forms that were defined by arrays of numbers. Although we are going to take the concrete approach, the concepts of abstract linear algebra will not go away. It is impossible to derive and analyze matrix algorithms without a knowledge of such things as subspaces, bases, dimension, and linear transformations. Consequently, after introducing vectors and matrices and describing how they combine, we will turn to the concepts of linear algebra. This inversion of the traditional order of presentation allows us to use the power of matrix methods to establish the basic results of linear algebra.