Page 200 - A Course in Linear Algebra with Applications

184 Chapter Six: Linear Transformations

Isomorphism theorems
There are certain theorems, known as isomorphism theo-
rems, which provide a link between linear transformations and
quotient spaces (which were defined in 5.3). Such theorems
occur frequently in algebra. The first theorem of this type is:
Theorem 6.3.7
IfT : V —+ W is a linear transformation between vector spaces
V and W, then

V/Kev(T) ~ Im(T).

Proof
Write K = Ker (T). We define a function S : V/K -> Im(T)
by the rule S{\ + K) = T(v). The first thing to notice is that
S is well-defined: indeed if u € K, then

T(v + u) = T(v) + T(u) = T(v) + 0 = T(v),

since T(u) = 0. Thus S(v + K) does not depend on the choice
of representative v of the coset v + K.
Next it is simple to verify that 5" is a linear transforma-
tion: for example,

^((v! + K) + (v 2 + K)) = S((vi + v 2 ) + K)
= T ( v i + v 2 )
= r(vi) + r(v ),
2
,
which equals S(v\ + K) + 5 (v 2 + K). In a similar way it can
be shown that S(c(v + K)) = cS(v + K).
Clearly the function S is surjective, so all we need do to
complete the proof is show it is injective. If S(y + K) = 0,
then T(v) = 0; thus v e K and v + K = 0 V/K- Hence, by
6.3.2, S is injective.