By Robert M. Thrall
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Extra resources for Vector Spaces and Matrices
Sample text
Chapter 10 presents the similarity theory with the usual canonical forms. It includes applications to geometry and to differential equations. Many further applications are considered in the exercises in this chapter. The final chapter on linear inequalities presents a brief introduction to this important topic, which has until now been neglected in elementary texts on matrix theory. We hope that what we have given will provide the student with sufficient background so that he can read books and papers on game theory and linear programming.
In arithmetic and elementary algebra we study operations on numbers and on symbols which represent numbers. In plane geometry we describe each point by a pair of numbers and so can bring the tools of algebra to bear on geometry. In solid geometry a triple of numbers is employed, and for higher dimensions we merely use more numbers to describe each point and proceed algebraically much as in the two- and three-dimensional cases. Even though for dimensions greater than three we have no direct physical representation for the geometry, it is still useful to employ geometric language in describing the algebraic results.
For n = 1 there is no product, and so g(a1) = a1 trivially. For n = 2 the only product is a1a2, and so g(a1, a2) = a1a2. For n = 3 there are only the two products a1a2a3 = (a1a2)a3 and a1(a2, a3), and the equality of these is just what the associative law states. Now make the induction hypothesis that the theorem is true for all products with less than r factors. Among the parentheses that define the product g(a1, , ar) consider only the outermost pair. They give where 1 s < r and g1, g2 are products with s, r − s factors, respectively.