Page 133 - Advanced Linear Algebra

Modules I: Basic Properties 117

Theorem 4.5 Let 4 and 5 be 9 -modules where 4 is free with basis
8 9-map ~¸ “ 0¹. Then we can define a unique ¢ 4 ¦ 5 by specifying
the values of arbitrarily for all 8 and then extending to 4  by
linearity, that is,

² # b Äb # ³ ~ # bÄb #

Homomorphisms
The term linear transformation is special to vector spaces. However, the
concept applies to most algebraic structures.

Definition Let 4 and 5 be 9 -modules. A function ¢ 4 ¦ 5 is an 9 -
homomorphism or 9 -map if it preserves the module operations, that is,

² " b #³ ~ ²"³ b ²#³
9
for all Á  9 and "Á #  4 . The set of all -homomorphisms from 4 to is
5
denoted by hom 9 ²4Á 5³ . The following terms are also employed:
)
1 An -endomorphism is an -homomorphism from 4 to itself.
9
9
)
2 An -monomorphism or -embedding is an injective -homomorphism.
9
9
9
3 An -epimorphism is a surjective -homomorphism.
)
9
9
4 An -isomorphism is a bijective -homomorphism.
)
9
9
It is easy to see that hom 9 ²4Á 5³ is itself an 9 -module under addition of
functions and scalar multiplication defined by

² ³²#³ ~ ² #³ ~ ² #³

Theorem 4.6 Let  ² hom 9 4 Á 5 ³ . The kernel and image of , defined as for

linear transformations by

ker² ³ ~¸#4 “ #~ ¹
and

im² ³ ~¸ #“#4¹
are submodules of 4 and 5 , respectively. Moreover, is a monomorphism if

and only if ker² ³ ~ ¸ ¹ .

If 5 is a submodule of the -module 4 , then the map ¢ 5 ¦ 4 defined by
9
²#³ ~ # is evidently an 9-monomorphism, called injection of 5 into 4.
Quotient Modules
The procedure for defining quotient modules is the same as that for defining
quotient vector spaces. We summarize in the following theorem.