30.10 Linear Mappings

Let mapp be a general mapping, with underlying relation F Í S ×R, where S and R are the source and the range of mapp, respectively. Then RespectsScalarMultiplication returns true if S and R are left modules with the left acting domain D of S contained in the left acting domain of R and such that (s,r) Î F implies (c \* s,c \* r) Î F for all c Î D, and false otherwise.

If mapp is single-valued then RespectsScalarMultiplication returns true if and only if the equation c * s^mapp = (c * s)^mapp holds for all c in D and s in S.

For a field F and a general mapping mapp, IsLinearMapping returns true if mapp is an F-linear mapping, and false otherwise.

A mapping f is a linear mapping (or vector space homomorphism) if the source and range are vector spaces over the same division ring D, and if f( a + b ) = f(a) + f(b) and f( s \* a ) = s \* f(a) hold for all elements a, b in the source of f and s Î D.

Let mapp be a general mapping. Then KernelOfMultiplicativeGeneralMapping returns the set of all elements in the source of mapp that have the identity of the range in their set of images.

(This is a monoid if mapp respects multiplication and one, and if the source of mapp is associative.)

Let mapp be a general mapping. Then CoKernelOfMultiplicativeGeneralMapping returns the set of all elements in the range of mapp that have the identity of the source in their set of preimages.

(This is a monoid if mapp respects multiplication and one, and if the range of mapp is associative.)

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