>>Yeah, sure. A vector space is a module over a field. That's absolutely >>true and probably given that way in Bourbaki somewhere, but even in >>selective institutions I can't believe students learn about modules >>before vector spaces. Is there really a source anywhere that pretends to >>_introduce_ vector spaces in terms of abelian groups?
>Er.. yeah. MacLane and Birkhoff, Algebra, 3rd ed. 1988.
>Chapter 5 Modules "An R-module is an additive abelian group together > with a function..."
>Chapter 6 Vector Spaces "A vector space is a module over a field F."
>I don't know if teachers actually use this as a textbook for an undergrad >class though.
It has happened. I first learned linear algebra from Herstein (himself, as well as his text "Topics in Algebra") at U of Chicago in 1969-70: "A nonempty set V is said to be a vector space over a field F if V is an abelian group ..."
Robert Israel email@example.com Department of Mathematics http://www.math.ubc.ca/~israel University of British Columbia Vancouver, BC, Canada V6T 1Z2