線形代数の教材について : 平行性を用いた2次の行列式および2次元ベクトルの内積の幾何学的構成

Abstract

This is a case study of teaching materials of the linear algebra. The linear algebra can be understood from two points of view, that is, algebraic and geometric. Here, we consider two elementary topics of the linear algebra; the determinants of order two and the inner products of two dimensional number vectors. The purpose of this paper is to study the geometric constructions of these objects. A parallelism is used for these constructions, more precisely, we use the segments calculation by D. Hilbert and a parallel projection along the line y = mx with slope m of a point to the x-axis or the y-axis. The determinant and inner product are constructed corresponding to the slope m = 1 and -1 respectively, since the determinant is alternative with respect to its columns and the inner product is symmetric bilinear form. As an application of the construction by using parallelism, we consider geometric constructions of the solution of the system of linear equations. It seems that the geometric construction by using parallelism is effective for intuitive understanding of the teaching materials stated above.