Linearity

Gerhard P. Hochschild
Additional contact information
Gerhard P. Hochschild: University of California, Department of Mathematics

Chapter Chapter IV in Perspectives of Elementary Mathematics, 1983, pp 36-49 from Springer

Abstract: Abstract Consider a commutative group V, whose composition map we indicate by +. Recall that the endomorphisms of V constitute a ring End(V). Now we suppose that there is given a ring homomorphism σ from some field F to End(V). The customary terminology for referring to this situation is to say that a makes V into a vector space over F. For a in F and υ in V, one abbreviates σ(α)(υ) by at;, and one calls σ(α) the scalar multiplication by a. The fact that σ(α) is an endomorphism of V is expressed by the formula $$ \alpha \left( {{v_1} + {v_2}} \right) = \alpha {v_1} + \alpha {v_2} $$ and the fact that σ is a ring homomorphism is expressed by the formulas $$ \left( {{\alpha_1} + {\alpha_2}} \right)v = {\alpha_1}v + {\alpha_2}v\left( {{\alpha_1}{\alpha_2}} \right)v = {\alpha_1}\left( {{\alpha_2}v} \right) $$

Keywords: Vector Space; Line Segment; Vector Space Versus; Ring Homomorphism; Determinant Function (search for similar items in EconPapers)
Date: 1983
References: Add references at CitEc
Citations:

There are no downloads for this item, see the EconPapers FAQ for hints about obtaining it.

Related works:
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: https://EconPapers.repec.org/RePEc:spr:sprchp:978-1-4612-5567-3_4

Ordering information: This item can be ordered from
http://www.springer.com/9781461255673

DOI: 10.1007/978-1-4612-5567-3_4

Access Statistics for this chapter

More chapters in Springer Books from Springer
Bibliographic data for series maintained by Sonal Shukla () and Springer Nature Abstracting and Indexing ().