Basis

Japanese: Basis

…One of the basic concepts in Marxist social theory and therefore historical theory. It is the opposite concept to the infrastructure (foundation). In the materialist view of history (historical materialism), the infrastructure is the totality of "the relations that are definite, necessary, and independent of people's will in the social production of life," that is, "the relations of production that correspond to a certain stage of development of the material productive forces." This "economic structure of society" is called infrastructure.

From [linear independence]

…When x ₁ , …, x _n are not linearly independent, they are said to be linearly dependent. When there are elements y ₁ , …, y _m of V , and any element x of V can be written as a linear combination of y ₁ , …, y _m , that is, x = β ₁ y ₁ + … + β _m y _m , V is said to be finite-dimensional, and when y ₁ , …, y _m are linearly independent, they are called a basis of V. A basis always exists, and the number m is constant regardless of the choice of basis. …

*Some of the terminology explanations that mention "Basis" are listed below.

Source | Heibonsha World Encyclopedia 2nd Edition | Information

Japanese:

…マルクス主義の社会理論ひいては歴史理論における基本概念の一つ。下部構造(土台)Basisと対概念をなす。唯物史観(史的唯物論)においては〈人々が生の社会的生産において入り込む一定の，必然的な，彼らの意思から独立な諸関係〉すなわち〈物質的な生産諸力の一定の発展段階に照応する生産諸関係〉の一総体，この〈社会の経済的構造〉を下部構造と呼ぶ。…

【一次独立】より

…x₁，……，x_nが一次独立でないとき，一次従属linearly dependentであるという。Vの元y₁，……，y_mがあって，Vの任意の元xがy₁，……，y_mの一次結合で書ける，すなわちx＝β₁y₁＋……＋β_my_mと表せるとき，Vは有限次元であるといい，さらにy₁，……，y_mが一次独立であるとき，それらをVの基底basisと呼ぶ。基底は必ず存在し，その数mは基底の選び方によらず一定である。…