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单词 ModulesAreAGeneralizationOfVectorSpaces
释义

modules are a generalization of vector spaces


A http://planetmath.org/node/1022module is the natural generalizationPlanetmathPlanetmath of a vector spaceMathworldPlanetmath, in fact, when working over a field it is just another word for a vector space.

If M and N are R-modules then a mapping f:MN is called an R-morphism (or homomorphismPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath) if:

x,yM:f(x+y)=f(x)+f(y)andxMλR:f(λx)=λf(x)

Note as mentioned in the beginning, if R is a field, these properties are the defining properties for a linear transformation.

Similarly in vector space terminology the image Imf:={f(x):xM} and kernelKerf:={xM:f(x)=0N} are called the range and null-space respectively.

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更新时间:2025/5/4 18:36:33